english
Archiwum seminariów
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09.02.43535 Marta Kwiatkowska University of Oxford |
Informatyka Teoretyczna Strategy synthesis for partially observable stochastic games with neural perception mechanisms |
Strategic reasoning is essential to ensure stable multi-agent coordination in complex environments, as it enables synthesis of optimal (or near-optimal) agent strategies and equilibria that guarantee expected outcomes, even in adversarial scenarios. Partially-observable stochastic games (POSGs) are a natural model for real-world settings involving multiple agents, uncertainty and partial information, but lack practical algorithms for computing or approximating optimal values and strategies. Recently, progress has been made for one-sided POSGs, a subclass of two-agent, zero-sum POSGs where only one agent has partial information while the other agent is assumed to have full knowledge of the state, with heuristic search value iteration (HSVI) proposed for computing approximately optimal values and strategies in one-sided POSGs.
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27.11.46268 Filip Konieczny |
Optymalizacja Kombinatoryczna Hall's Theorem for hypergraps |
Standard Hall's Theorem provides sufficient and necessary condition for a bipartite graph to have a perfect matching. We will formulate and proof generalization of this theorem in hypergraph setting. Interestingly, we make use of Sperner lemma and approach the problem from topological perspective.
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24.03.46281 Rafał Pyzik |
Optymalizacja Kombinatoryczna The Alon-Tarsi number of K3,3-minor-free graphs |
For a graph G, denote AT(G) as the Alon-Tarsi number of G. It is known, that if a graph G is planar then AT(G) ≤ 5, AT(G - M) ≤ 4 for some matching M in G and AT(G - F) ≤ 3 for some forest F in G. Also, by Wagner's theorem, G is planar if and only if it doesn't contain K5 and K3,3 as a minor. We prove these three Alon-Tarsi number bounds for G that is only K3,3-minor-free, strengthening the result for planar graphs.
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21.10.81865 Alexander Wolff Universität Würzburg |
Informatyka Teoretyczna Eliminating Crossings in Ordered Graphs |
Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with n vertices and m edges can be computed in 2m⋅poly(n) time. An O(log n)-approximation of this number can be computed efficiently. We can decide in 2O(d√k⋅log(d+k))⋅poly(n) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h=1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h>1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. Joint work with Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, and Yushi Uno. |
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14.02.16156 Maria Chudnovsky Princeton University |
Informatyka Teoretyczna TBA - 2024.06.05 - Maria Chudnovsky |
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02.12.18889 Jakub Fedak |
Optymalizacja Kombinatoryczna TBA Jakub Fedak |
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30.03.18902 Jędrzej Hodor |
Optymalizacja Kombinatoryczna TBA Jędrzej Hodor |
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09.04.38055 Łukasz Gniecki |
Optymalizacja Kombinatoryczna TBA Łukasz Gniecki |
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18.11.27115 Aleksander Katan |
Optymalizacja Kombinatoryczna Dynamic monopolies in directed graphs: The spread of unilateral influence in social networks |
Given a digraph G = (V, E), a threshold function t:V→N, and a subset D0 of V, we define the procedure of the influence spread as follows: in turn i, a vertex v joins Di if it either is in Di-1, or has at least t(v) in-neighbors in Di-1. A subset D of V is called a t-dynamic monopoly if there exists a turn j such that Dj = V. We will discuss the hardness of finding the smallest t-dynamic monopoly when t(v) = 2, as well as prove that every graph admits a t-dynamic monopoly of size |V|/2 when t(v) = ⌈(degin(v)+1)/2⌉. |
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24.07.27103 Ignacy Buczek |
Optymalizacja Kombinatoryczna Flip-width and its basic properties |
Some of the recent developments in structural graph theory revolve around the idea of generalizing notions that have proven successful for sparse graphs to more general settings. In that line of work, we present a proposition of a new graph parameter, called flip-width, for which we present strong proof that it could serve as a viable generalization of several notions in the sparsity theory. Most notably, we conjecture it coincides with the class of graphs for which model checking is fixed-parameter tractable.
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04.10.24369 Michał Pilipczuk University of Warsaw |
Informatyka Teoretyczna Minor testing in almost linear time |
For every fixed graph H, we give an algorithm that given a graph G with m edges, tests whether H is a minor of G in time OH(m1+o(1)). This improves the classic cubic-time algorithm of Robertson and Seymour, and its improvement to quadratic time by Kawarabayashi, Kobayashi, and Reed. By the Graph Minors Theorem, we obtain, as a corollary, an OC(m1+o(1))-time membership test for every minor-closed class of graphs C. Further, the algorithm also works for the rooted version of the problem, so it can be used to solve the Disjoint Paths problem in time Ok(m1+o(1)). This is a joint work with Tuukka Korhonen and Giannos Stamoulis |
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23.04.70922 Sebastian Spyrzewski |
Optymalizacja Kombinatoryczna Mapping Matchings to Minimum Vertex Covers: Kőnig's Theorem Revisited |
It is a very well-known result that the size of maximum matching is equal to the size of a minimum vertex cover. Kőnig’s proof of this fact gave an algorithm for finding a minimum vertex cover from a maximum matching. In this paper, we revisit this algorithm and see how it implies the connection between the two types of structures.
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27.12.70909 Katarzyna Kępińska |
Optymalizacja Kombinatoryczna Two results on layered pathwidth and linear layouts |
In this paper, we study the relations of layered pathwidth to other graph parameters. In particular, we show that the stack number of G is at most 4 times the layered pathwidth of G. The Second result bounds layered pathwidth for graphs with track number at most 3.
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09.03.68176 Bartosz Walczak Jagiellonian University |
Informatyka Teoretyczna Polynomial-time recognition and maximum independent set in Burling graphs |
A Burling graph is an induced subgraph of some graph in Burling's construction of triangle-free high-chromatic graphs. Burling graphs can also be characterized as graphs with so-called strict frame representations, i.e., intersection models by suitably restricted rectangular frames. We provide a polynomial-time algorithm which decides whether a given graph is a Burling graph and if it is, constructs its strict frame representation. We also provide a polynomial-time algorithm for the maximum independent set problem in Burling graphs. This establishes Burling graphs as the first known hereditary class of graphs that admits such an algorithm and is not χ-bounded, answering a question of Thomassé, Trotignon, and Vušković.
Joint work with Paweł Rzążewski. |
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16.12.51756 Jan Klimczak |
Optymalizacja Kombinatoryczna 3-Colorability of 4-Regular Hamiltonian Graphs |
It is a well-known result known as the 'cycle-plus-triangles' theorem that every graph on 3n vertices consisting of a Hamiltonian cycle and n node-disjoint triangles is 3-colorable. To improve this result we search for less strict restrictions of G\H. By reduction we show that two following problems are NP-complete: (1) 3-colorability of 4-regular Hamiltonian graphs. (2) 3-colorability of 4-regular Hamiltonian graphs whose inner cycles (connected components after Hamiltonian cycle removal) are non-selfcrossing k-gons.
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21.08.51744 Maksym Zub |
Optymalizacja Kombinatoryczna Recisitng Tucker's Algorithm To Color Circular Arc Graphs |
The circular arc coloring problem consists of finding a minimum coloring of a circular arc family F such that no two intersecting arcs share a color. Let l be the minimum number of circular arcs in F that are needed to cover the circle. Tucker shows in [SIAM J. Appl. Math., 29 (1975), pp. 493–502], that if l ≥ 4, then ⌊3/2L⌋ colors suffice to color F, where L denotes the load of F. We extend Tucker’s result by showing that if l ≥ 5, then ⌈(L-1)/(L-2)L⌉ colors suffice to color F, and this upper bound is tight. |
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03.11.49010 Hoang La Université Paris-Saclay |
Informatyka Teoretyczna Graph reconstruction with connectivity queries |
We study a problem of reconstruction of connected graphs where the input gives all subsets of size k that induce a connected subgraph. Originally introduced by Bastide et al. (WG 2023) for triples (k=3), this problem received comprehensive attention in their work, alongside a study by Qi, who provided a complete characterization of graphs uniquely reconstructible via their connected triples, i.e. no other graphs share the same set of connected triples. Our contribution consists in output-polynomial time algorithms that enumerate every triangle-free graph (resp. every graph with bounded maximum degree) that is consistent with a specified set of connected k-sets. Notably, we prove that triangle-free graphs are uniquely reconstructible, while graphs with bounded maximum degree that are consistent with the same k-sets share a substantial common structure, differing only locally. We suspect that the problem is NP-hard in general and provide a NP-hardness proof for a variant where the connectivity is specified for only some k-sets (with k at least 4). |
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11.08.32591 Mateusz Hurkała |
Optymalizacja Kombinatoryczna The two possible values of the chromatic number of a random graph |
Given d in range (0, ∞) let kd be the smallest integer such that d < 2k log k. We prove that the chromatic number of a random graph G(n,d/n) is either kd or kd+1 almost surely (probability thends to 1 as n approches ∞). And If d in range (2k log k - log k, 2k log k) we further prove that the chromatic number almost surely equals k+1.
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16.04.32579 Rafał Kajca |
Optymalizacja Kombinatoryczna Circular-arc graph coloring and unrolling |
The register periodic allocation problem is viewed as unrolling and coloring the underlying structure of circular-arc graph. The problem is to find relations between the unrolling degree and the chromatic number. For this purpose we distinguish cyclic colorings that can be found by means of the meeting graph and non-cyclic ones for which we prove the asymptotic property: let r be the width of the original interval family. Then the u-unrolled graph is r or (r+1)-colorable for u large enough. |
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28.06.29845 Jean Cardinal Université Libre de Bruxelles |
Informatyka Teoretyczna A rectangulation is a decomposition of a rectangle into finitely many rectangles |
Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. |
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06.04.13426 Filip Jasionowicz |
Optymalizacja Kombinatoryczna The Lefthanded Local Lemma Characterizes Chordal Dependency Graphs |
Shearer characterized the family L of dependency graphs labeled with probabilities such that for every family of events that can be modeled with a graph from L there is a positive probability that none of the events from this family occur. The authors show that unlike the Lovász Local Lemma (which is less powerful than the Shearer's condition on every nonempty graph) a lefthanded variant of LLL is equivalent to Shearer's condition for all chordal graphs. This leads to simple algorithm to determine whether a given label chordal graph is in L.
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10.12.13413 Agata Margas |
Optymalizacja Kombinatoryczna Euler circuits and DNA sequencing by hybridization |
In the problem of DNA sequencing by hybridization, it is useful to know the number of possible reconstructions of a long DNA string given known short substrings. This number is determined by the pattern of repeated substrings, and in the pattern considered here each substring occurs at most twice. A pairing is a sequence in which each symbol occurs exactly twice, and each pairing induces a 2-in, 2-out graph. We will count the number of pairings of given length, for which the induced graph has exactly k Euler circuits. |
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21.02.10680 David Conlon California Institute of Technology |
Informatyka Teoretyczna Additive combinatorics without (much) addition |
We describe recent progress on a number of related themes in additive combinatorics, including estimating the clique number of random Cayley graphs and showing that there are Cayley graphs which are good Ramsey graphs. Surprisingly, the proofs of these results rely only weakly on the group structure and the proofs are mainly about the structure of properly edge-coloured graphs. Joint work with Jacob Fox, Huy Tuan Pham and Liana Yepremyan. |
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28.10.76389 Clément Rambaud Université Côte d'Azur |
Informatyka Teoretyczna Inversions in oriented graphs |
The inversion of a set X of vertices in an oriented graph consists in reversing the direction of all arcs of the subdigraph induced by X. This generalization of single arc reversal introduced by Belkhechine et al. yields a notion of distance between orientations of a same graph where two orientations are at distance one if and only if there is a set X whose inversion transforms one into the other. First we will discuss the minimum number of inversions required to make an oriented graph acyclic, and in particular a proof of the existence of n-vertex oriented graphs at distance n-o(n) of any acyclic orientation. We also investigate the minimum number of inversions needed to make an oriented graph k-strongly-connected, especially in the case of tournaments. Finally, we show various bounds on the maximum distance between two orientations of a same graph. This gives an undirected graph parameter that we show to be tied to several known parameters, including the star chromatic number and acyclic chromatic number. We also prove that two orientations of a same planar graph are at distance at most 12. Most of these results rely on an algebraic point of view that allows us to use linear algebra over the field with two elements. This is joint work with Guillaume Aubian, Julien Duron, Frédéric Havet, Florian Horsch, Felix Klingelhoefer, Nicolas Nisse, and Quentin Vermande. |
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06.08.59970 Kamil Galewski |
Optymalizacja Kombinatoryczna Shannon Entropy of Ramsey Graphs with up to Six Vertices |
The Ramsey theorem asserts that for sufficiently large complete graphs, where edges are colored with a fixed number of colors, there must exist a monochromatic clique of a predetermined size. In this presentation, I will introduce a method for computing the Shannon entropy of bi-colored Ramsey complete graphs, demonstrated through examples of graphs containing up to six vertices. |
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11.04.59958 Piotr Kaliciak |
Optymalizacja Kombinatoryczna An algorithm for identifying cycle-plus-triangles graphs |
A cycle-plus-triangle graph is a union of node-disjoint triangles and a Hamiltonian cycle. It is known that such graphs are 3-colorable, but the problem of finding any 3-coloring is still open. The authors show a polynomial time algorithm for deciding if a given graph is cycle-plus-triangle, and if this is the case, the algorithm provides the decomposition into the triangles and a cycle. |
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22.06.57224 Gábor Tardos Alfréd Rényi Institute of Mathematics |
Informatyka Teoretyczna Forbidden acyclic patterns in 0-1 matrices |
A zero-one matrix M is said to contain another zero-one matrix A if we can delete some rows and columns of M and replace some 1-entries with 0-entries such that the resulting matrix is A. The extremal function of A, denoted ex(n,A), is the maximum number of 1-entries that an n×n zero-one matrix can have without containing A. The systematic study of this function for various patterns A goes back to the work of Füredi and Hajnal from 1992. The field has many connections to other areas such as classical Turán type extremal graph theory and Davenport-Schinzel theory and has many applications in mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices. Füredi and Hajnal conjectured an O(n·log n) bound for them, while Pach and Tardos conjectured a weaker n·polylog n bound. Pettie refuted the stronger conjecture with an acyclic pattern whose extremal function he showed to be Ω(n·log n · loglog n).
In a recent joint work with Seth Pettie we found the extremal function ex(n,Ak) asymptotically for certain simple 2×k acyclic patterns to be Θ(n·(log n/loglog n)k−2) for k>3. This shows that the Pach-Tardos conjecture is tight if true. The conjecture itself is still wide open. |
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31.03.40805 Michał Mizołek |
Optymalizacja Kombinatoryczna An O(n^2) algorithm for coloring proper circular arc graphs |
A graph qualifies as a circular arc graph when every node corresponds to an arc on a circle, with adjacency between any two nodes depending on the overlapping of their respective arcs. For a circular arc graph to be considered proper, it requires to be characterized by the unique property that no arc fully encloses another within its bounds. The paper introduce an efficient algorithm with a quadratic time complexity, designed to evaluate the colorability of these graphs, addressing the challenge of assigning k distinct colors to a graph with n vertices without violating adjacency constraints. The significance of this work lies in its contribution to graph theory, providing a an approach to understanding the structural properties of proper circular arc graphs and their implications for graph coloring problems. |
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04.12.40792 Maciej Sanocki |
Optymalizacja Kombinatoryczna Randomized Primal-Dual analysis of RANKING for Online Bipartite Matching |
The online bipartite matching problem focuses on finding a matching of the greatest cardinality for the bipartite graph, while vertices and their sets of edges from the „right side” are revealed one by one. The algorithm has to decide on, which edge to include in matching immediately upon the arrival. This paper shows, yet again, a proof for 1-1/e competitiveness of RANKING algorithm for the online bipartite matching problem. This time however, the proof is via a randomised primal-dual argument. |
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30.07.21627 Bartłomiej Błoniarz |
Optymalizacja Kombinatoryczna Cooperative colorings of trees and of bipartite graphs |
In a system of graphs (G1, ..., Gm) sharing the same set of vertices (V), a cooperative coloring involves selecting vertex sets (I1, ..., Im) where each Ij is independent in Gj, and the union of all sets equals V. For a graph class C and integer d we are concerned with the minimum m, such that every m graphs in this class with maximum degree d, can be cooperatively colored. The paper shows bounds on the value of m for trees and bipartite graphs. |
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10.10.18893 Jacob Fox Stanford University |
Informatyka Teoretyczna Structure theorems for intersection patterns of geometric objects |
In this talk we discuss Szemerédi-type structure theorems, Ramsey-type theorems, and Turán-type theorems for intersection patterns of geometric objects and their relationships to each other. In particular, we discuss recent such results on intersection graphs of pseudo-segments and an application which gives a new upper bound on the number of edges of a simple topological graph with no k pairwise disjoint edges. Joint work with János Pach and Andrew Suk. |
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14.06.81861 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Some open problems from combinatorics and algorithmics |
Some open problems and the latest results regarding majority coloring are presented. In addition, hypotheses were put forward regarding the hat guessing number. |
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01.05.70945 Filip Jasionowicz |
Optymalizacja Kombinatoryczna Four Pages Are Indeed Necessary for Planar Graphs |
An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by demonstrating planar graphs that require four pages in any of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.
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22.01.70922 Artemy Oueiski |
Optymalizacja Kombinatoryczna A simple linear time algorithm for cograph recognition |
Cographs are precisely the P4-free graphs. It is shown that a cograph can be uniquely represented by a special tree, called a cotree, where the leaves of the cotree correspond to the vertices of the cograph. An algorithm for recognizing cographs is considered, operating in linear time through two steps. In the first step partition refinement is used to create a factorizing permutation. At the second step, the permutation is tested to verify whether the graph is a cograph. Then algorithms for deriving the characteristics (pathwidth, treewidth, number of cliques) of a cograph from its cotree are explored.
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27.09.70909 Katzper Michno |
Optymalizacja Kombinatoryczna Another approach to non-repetitive colorings of graphs of bounded degree |
A non-repetitive graph coloring (of vertices or edges) is a coloring such that all sequences of colors induced by paths in the graph are non-repetitive (square-free), which means that they do not contain any consecutive subsequence that is a square. The non-repetitive number of a graph is the minimal number of colors in a non-repetitive vertex coloring (resp. non-repetitive index for coloring edges). There are also list counterparts of these numbers. Many maximal degree related upper bounds for non-repetitive number (resp. index) have been established commonly using the Lovász Local Lemma or entropy compression method. The author of this paper introduces another method of proving these bounds, which is closely related to the entropy compression method, but generates simpler and more elementary proofs. The author provides some minor improvements to non-repetitive number in several cases and matches some of already known bounds using the new technique. |
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09.12.68175 Sergio Cabello University of Ljubljana and IMFM, Slovenia |
Informatyka Teoretyczna Packing d-dimensional balls into a (d+1)-dimensional container |
We consider the problems of finding in d+1 dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of d-dimensional unit-radius balls can be packed under translations. We give a constant-factor approximation algorithm for each of these containers. We also show that for n such balls, a container of volume O(n1−1/d) is always sufficient and sometimes necessary. Joint work with Helmut Alt, Otfried Cheong, Ji-won Park and Nadja Seiferth. |
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16.09.51756 Milana Kananovich |
Optymalizacja Kombinatoryczna A Linear Recognition Algorithm for Cographs. A Simple Linear Time LexBFS Cograph Recognition Algorithm. |
Cographs are the graphs formed from a single vertex under the closure of the operations of union and complement. Another characterization of cographs is that they are undirected graphs with no induced paths on four vertices. Cographs have a unique tree representation called a cotree. We consider two linear time algorithms for recognizing cographs and constructing their cotree representation (or the reason why it is not a cograph, the 2nd algorithm gives us P4): a step-by-step recognition algorithm (where we have a list of conditions that must not be violated for the cograph) and LexBFS recognition algorithm (it uses a LexBFS approach, and introduces a new variant of LexBFS, operating on the complement of the given graph G and breaking ties concerning an initial LexBFS).
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22.05.51744 Sebastian Spyrzewski |
Optymalizacja Kombinatoryczna List coloring with requests |
In this paper we consider the problem of L-coloring graph G with the given list assignment L, but with additional requests giving the preferred color of some vertices. We ask a question of how many of these preferences can be respected while L-coloring G. We present a connection between weighted and unweighted requests and show that for degenerate graphs there is always a constant fraction of preferences that can be satisfied. |
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04.08.49010 Jim Geelen University of Waterloo |
Informatyka Teoretyczna Average plane size |
Consider a finite set of distinct points in d-dimensional Euclidean space. A line is said to be spanned if it contains two distinct points from the given set, and a plane is spanned if it contains three non-collinear points from the given set. In 1941, Melchior proved that the average number of given points on a spanned line is bounded above by 3, unless the given points all lie on a single line. We prove that the average number of given points on a spanned plane is bounded above by an absolute constant, unless all of the given points lie on a single plane or they lie on the union of two lines. This is joint work with Rutger Campbell and Matthew Kroeker. |
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12.05.32591 Aleksander Katan |
Optymalizacja Kombinatoryczna Countable graphs are majority 3-choosable |
A majority coloring of a graph is a vertex coloring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. The Unfriendly Partition Conjecture states that every countable graph admits a majority 2-coloring. It is known that for every (not necessarily countable) graph a majority 3-coloring always exists. Anholcer, Bosek, and Grytczuk have recently proven that every countable graph is majority 4-choosable, and we will see an improvement of this result to lists of size 3, as well as a simplified version of the proof that countable graphs are 3-colorable. |
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15.01.32579 Łukasz Gniecki |
Optymalizacja Kombinatoryczna The Alon Tarsi Number of Planar Graphs - a Simple Proof |
The Alon-Tarsi number of a Graph, AT(G), is a value defined by considering eulerian subsets of edges of a chosen orientation of the graph. It has many connections to the domain of graph coloring. For example, the choice number of a graph, ch(G), is bounded by the Alon-Tarsi number, AT(G). In this paper, we will see a simple proof, in the style of Thomassen's induction, of the statement that for any planar graph G, AT(G) is at most 5. Alongside, we will see that any planar G has a matching M, such that AT(G - M) is at most 4. |
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29.03.29845 Peter Allen London School of Economics and Political Science |
Informatyka Teoretyczna Universality for degenerate graphs |
A graph G is universal for a family ℱ of graphs if for each F in ℱ there is a copy of F in G (not necessarily induced, and the copies are not necessarily disjoint). Alon and Capalbo considered the case that ℱ is the family of n-vertex graphs with maximum degree k, and showed that there is a universal graph for this family with O(n2-2/k) edges, which is sharp. Alon asked what the answer is if one replaces 'maximum degree' with 'degeneracy'. We give a probabilistic construction of a universal graph for the family of n-vertex d-degenerate graphs with Õ(n2-1/d) edges, which is optimal up to the polylog. In this talk, I will describe the construction and give most of the details of the proof of its universality. This is joint work with Julia Boettcher and Anita Liebenau. |
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07.05.59970 Kamil Galewski |
Optymalizacja Kombinatoryczna On two generalizations of the Alon–Tarsi polynomial method |
The Alon-Tarsi number of a graph G=([n], E) is the smallest integer k, such that there exists a monomial x1d1x2d2...xndn in the expansion of the graph polynomial of G having non-zero coefficient and satisfying di < k for all i∈[n]. Using Combinatorial Nullstellensatz, one can show that this number is an upper bound on the choice number of the graph (and thus on its chromatic number). Alon and Tarsi presented a way of checking non-zeroness of the coefficient of the monomial x1d1...xndn in case di = outdegD(i) for some orientation D of graph G - it is sufficient to check whether the difference between the number of the odd and even Eulerian suborientations of D is non-zero. In this presentation, I will show a generalization of this result - for each D, there exists an infinite family of functions f mapping suborientations of D to real numbers, such that the coefficient mentioned above is non-zero iff the sum of f(A) over all the suborientations A of D is non-zero. |
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10.01.59958 Ignacy Buczek |
Optymalizacja Kombinatoryczna A note on computer-assisted proofs in flag algebras |
With the help of CSDP solvers, one can use computer assistance to obtain correct proofs in flag algebras. In the most common implementations, the programs return the best possible bound on the objective function, together with some information on the extremal graphon. However, for more complicated graphons, this information is usually insufficient to fully retrieve the extremal graphon. We present how one can gather more information on the extremal graphon by adding temporary assumptions to the program, using a non-trivial example that we stumbled upon in our unpublished work on balanced bipartitions of K4-free graphs. |
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23.03.57224 Paweł Rzążewski Warsaw University of Technology |
Informatyka Teoretyczna Understanding graphs with no long claws |
A classic result of Alekseev asserts that for connected H the MAX INDEPENDENT SET (MIS) problem in H-free graphs in NP-hard unless H is a path or a subdivided claw. Recently we have witnessed some great progress in understanding the complexity of MIS in Pt-free graphs. The situation for forbidden subdivided claws is, however, much less understood. During the talk we will present some recent advances in understanding the structure of graphs with no long induced claws. We are able to use them to obtain a quasipolynomial-time algorithm for the problem. |
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04.09.40792 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Wythoff's game |
Consider an n×m chessboard with a single queen placed somewhere. There are two players and in order to win, one has to place the queen in the left-bottom corner. A player can either move the queen diagonally towards the left-bottom or vertically towards the left or bottom. It turns out that sometimes the first player has a winning strategy and sometimes the second player. The characterization is mathematically beautiful. The first player has a winning strategy if and only if there is a non-negative integer n such that the queen starts in the position (⌊nφ⌋, ⌊nφ2⌋), where φ is the golden ratio. |
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30.04.21627 Piotr Kaliciak |
Optymalizacja Kombinatoryczna Hat guessing numbers of strongly degenerate graphs |
Consider a game with n players, each placed on one of the vertices of graph G. Each player is given a hat, but they cannot see their hat color; they can only see the colors of the hats worn by their neighbors in G. The objective for the players is to ensure that at least one player correctly guesses the color of their hat. The hat guessing number of graph G, denoted by HG(G), is the maximum number of colors for which the players possess a winning strategy. We present an upper bound for the hat guessing number of d-degenerate and outerplanar graphs. |
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11.07.18893 Paul Bastide LaBRI, Bordeaux |
Informatyka Teoretyczna Skipless chain decompositions and improved poset saturation bounds |
We show that given m disjoint chains in the Boolean lattice, we can create m disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer two conjectures about the asymptotics of induced saturation numbers for the antichain, which are defined as follows. For positive integers k and n, a family F of subsets of {1,...,n} is k-antichain saturated if it does not contain an antichain of size k (as induced subposet), but adding any set to F creates an antichain of size k. We use sat*(n,k) to denote the smallest size of such a family. With more work we pinpoint the exact value of sat*(n,k), for all k and sufficiently large n. Previously, exact values for sat*(n,k) were only known for k up to 6. We also show that for any poset P, its induced saturation number (which is defined similar as for the antichain) grows at most polynomially: sat*(n,P)=O(nc), where c≤|P|2/4+1. This is based on joint works with Carla Groenland, Maria-Romina Ivan, Hugo Jacob and Tom Johnston. |
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06.06.84611 Jan Klimczak |
Optymalizacja Kombinatoryczna On the equitable distribution of points on the circle |
The stick-breaking problem is equivalent to the online resource allocation problem, where we possess one unit of resource and we want to fairly distribute fractions of it between people, whose number is unknown at the beginning and upon person's arrival we are only allowed to decrease the share of resource of one person and transfer it to the newcomer. We present various solutions to this problem and analyze their efficiency. |
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08.02.84599 Rafał Pyzik |
Optymalizacja Kombinatoryczna Online Algorithms for Maximum Cardinality Matching with Edge Arrivals |
In the edge arrival model for the online maximum matching problem, edges are sequentially presented and each of them is accepted for the final matching or discarded. We present the Min-Index framework - a family of randomized algorithms for this problem. Using this framework, we show a 5/9-competitive algorithm when the graph is a tree. Moreover, we show that any algorithm in the edge arrival model is at most 0.5914 competitive. |
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22.04.81865 Matthieu Rosenfeld LIRMM, Montpellier |
Informatyka Teoretyczna A simple counting argument applied to graph colorings |
The Lovász Local Lemma is one of the central tools of Erdős' probabilistic method. This rather simple lemma has been applied to SAT formulas, graph colorings, hypergraph coloring, combinatorics on words, geometry, and countless other topics. This Lemma essentially tells that given a set of "bad events", under the right conditions, the probability that no events occur is nonzero. It implies the existence of a coloring or an affection of the variables with the desired properties. There are many versions of the Lovász Local Lemma that apply to different situations. Many related techniques that apply to similar problems have emerged in the last 20 years (entropy compression, cluster expansion, local cut lemma...). Recently, I have introduced a counting argument that belongs to the same family of technique. The main interest of this counting argument is that it is really simple to use and it has already been applied to different problems by a few different authors. |
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29.01.65446 Izabela Tylek |
Optymalizacja Kombinatoryczna Any 7-chromatic graph has a K7 or K4,4 as a minor |
One of perhaps the most important and interesting unsolved problems in the field of graph theory is the Hadwiger conjecture, which states that every k-chromatic graph has a Kk-minor. It has been proven to be true for k≤6; the cases k=5 and k=6 have been shown to be equivalent to the four-color theorem. The conjecture remains unsolved for k≥7, but some partial results are known. We will look closer at one of them, showing that any 7-chromatic graph has a K7 or K4,4 as a minor. |
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04.10.65433 Justyna Jaworska |
Optymalizacja Kombinatoryczna An O(n√n) algorithm to color proper circular arcs |
A proper circular arc family F is a set of arcs on a circle such that no arc is contained within another. We consider incidence graphs for such arc families. On proper circular-arc graphs, the coloring problem is polynomially solvable, most recently, in O(n1.5 log n) time (Teng and Tucker). It's due to the fact that the (q-colorability) problem can be reduced to a shortest path problem. In this note, we improve Teng and Tucker’s algorithm obtaining O(n1.5) running time. |
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16.12.62699 Gábor Damásdi Alfréd Rényi Institute of Mathematics |
Informatyka Teoretyczna Monochromatic configurations on the circle |
In this lecture we will discuss a surprising combinatorial conjecture. For k≥3 call a k-tuple (d1,d2,...,dk) with d1≥d2≥...≥dk>0 and d1+d2+...+dk=1 a Ramsey k-tuple if the following is true: in every two-colouring of the circle of unit perimeter, there is a monochromatic k-tuple of points in which the distances of cyclically consecutive points, measured along the arcs, are d1,d2,...,dk in some order. By a conjecture of Stromquist, if di=2k-i/(2k-1), then d1,d2,...,dk is Ramsey. Using Sat solvers we showed that the conjecture holds for k≤7. Our main result is a proof of the converse of this conjecture. That is, we show that if (d1,d2,...,dk) is Ramsey, then di=2k-i/(2k-1). We do this by finding connections of the problem to certain questions from number theory about partitioning N into so-called Beatty sequences.
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23.09.46280 Maciej Sanocki |
Optymalizacja Kombinatoryczna Two-sided Online Bipartite Matching and Vertex Cover: Beating the Greedy Algorithm |
In the original setting of online bipartite matching, vertices from only one side of the bipartite graph are online. This time however we will focus on generalization, in which all vertices can be online. An algorithm for it should maintain a b-matching and try to maximize its size. We show that this problem can be attacked by considering the complementary “dual” problem, a two-sided online bipartite vertex cover. |
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29.05.46268 Katarzyna Kępińska |
Optymalizacja Kombinatoryczna On Two problems of Defective Choosability of Graphs |
Graph G is (k,d,p)-choosable if given list assignment L where |L(v)| is at least k for each vertex v and the number of all available colors is p, there exists L-coloring such that maximum degree of monochromatic subgraph is at most d. This paper shows two constructions of graphs: 1-defective 3-choosable that are not 4-choosable and (k,d,l)-choosable that are not (k,d,l+1)-choosable. |
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11.08.43534 Piotr Micek Jagiellonian |
Informatyka Teoretyczna Tight bound for the Erdős-Pósa property of tree minors |
Let T be a tree on t vertices. We prove that for every positive integer k and every graph G, either G contains k pairwise vertex-disjoint subgraphs each having a T minor, or there exists a set X of at most t(k-1) vertices of G such that G-X has no T minor. The bound on the size of X is best possible and improves on an earlier f(t)k bound proved by Fiorini, Joret, and Wood (2013) with some very fast growing function f(t). Our proof is moreover very short and simple. Joint work with Vida Dujmović, Gwenaël Joret, and Pat Morin |
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23.01.27103 Karolina Okrasa Warsaw University of Technology |
Optymalizacja Kombinatoryczna Graph Homomorphisms: From Structure to Algorithms |
For two graphs G and H, a homomorphism from G to H is a function that maps the vertices of G to the vertices of H in a way that edges are preserved. Graph homomorphisms are a generalization of graph colorings: if H is a complete graph on k vertices, then homomorphisms from G to H are precisely the k-colorings of G and vice versa. It seems natural to follow the lines of research for the coloring problem to study the more general homomorphism problem. In the talk, I will focus on determining the complexity of the homomorphism problem (and its list variant) when we assume the class of input instances is somehow restricted, e.g., by bounding some structural parameter of an instance, or excluding the instances that contain some fixed graph as an induced subgraph. We examine to which extent the variety of tools developed to work on coloring problems can be applied, and show more general methods to approach these problems. |
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05.04.24369 Torsten Ueckerdt Karlsruhe Institute of Technology |
Informatyka Teoretyczna When Surrounding is not Catching in Cops and Robber |
After a short introduction of the classical game of Cops and Robber on graphs, we shall discuss two recently introduced variants in which the robber only loses when he is completely surrounded by the cops. In the first variant the robber is surrounded when he sits at a vertex v and there is at least one cop on each neighbor of v. In the second variant cops occupy edges of the graph and the robber (still moving on vertices) is surrounded if he sits at a vertex v and there is at least one cop on each incident edge at v. We shall compare these games with each other and also with the classical game in which the robber is already caught when one cop sits on the same vertex as the robber. |
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18.09.73659 Agata Margas |
Optymalizacja Kombinatoryczna Making the Rules of Sports Fairer |
The rules of many sports are not fair - they do not ensure that equally skilled competitors have the same probability of winning. As an example, the penalty shootout in soccer, wherein a coin toss determines which team kicks first on all five penalty kicks, gives a substantial advantage to the first-kicking team, both in theory and in practice. We show that a so-called Catch-Up Rule for determining the order of kicking would not only make the shootout fairer but is also essentially strategyproof. By contrast, the so-called Standard Rule now used for the tiebreaker in tennis is fair. |
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24.05.73647 Mikołaj Kot |
Optymalizacja Kombinatoryczna Circle graphs and monadic second-order logic |
Circle graph is intersection graph of set of chords od a circle. Such set is called chord diagram. It can also be described by word with two occurrences of each letter. If given graph is prime for the split decomposition, it has unique representation as chord diagram, and this diagram can be defined by monadic second-order formulas with even cardinality set predicate. The article also states that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width. |
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28.09.73596 Jan Klimczak, Szymon Wojtulewicz |
Approximating Knapsack and Partition via Dense Subset Sums |
| Kwestia złożoności (1 - ε)-aproksymacji problemu plecakowego i problemu podziału pozostaje nierozstrzygnięta. Prezentujemy algorytmy: - (1 - ε)-aproksymacja problemu plecakowego w złożoności O(n + (1/ε)^(2.2)) - (1 - ε)-aproksymacja problemu podziału w złożoności O(n + (1/ε)^(1.25)) Obie techniki wykorzystują poprzednie rezultaty na temat konwolucji gęstych zbiorów. Wykorzystane zostały też nowe sposoby przyspieszenia metody 'dziel i zwyciężaj', która jest często wykorzystywana w problemach addytywnych. |
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05.08.70913 Torsten Mütze University of Warwick |
Informatyka Teoretyczna A book proof of the middle levels theorem |
In this lecture I present an elementary and fully self-contained proof of the middle levels conjecture (now theorem), which asserts that the subgraph of the (2n+1)-dimensional hypercube induced by all bitstrings with Hamming weight n or n+1 admits a Hamilton cycle for every n≥1. |
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13.05.54494 Maksym Zub |
Optymalizacja Kombinatoryczna A note concerning the Grundy and b-chromatic number of graphs |
The Grundy number Γ(G) is the maximum number of colors used by the First-Fit coloring of G denoted by Γ(G). Similarly, the b- chromatic number b(G) of G expresses the worst-case behavior of another well-known coloring procedure i.e. color-dominating coloring of G. We obtain some families of graphs F for which there exists a function f(x) such that Γ(G) ≤ f(b(G)), for each graph G from the family. Call any such family (Γ,b)-bounded family. We conjecture that the family of b-monotone graphs is (Γ, b)-bounded and validate the conjecture for some families of graphs. |
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16.01.54482 Jakub Fedak |
Optymalizacja Kombinatoryczna The complexity of coloring circular arcs and chords |
Numerous graph problems, known to be NP-complete, become polynomial when restricted to specific graph types, such as planar graphs or comparability graphs. The article shows the NP-completeness of graph coloring for circular-arc graphs and circle graphs, as well as a polynomial algorithm for producing a K-coloring for circular-arc graphs if one exists. To prove the NP-completeness of graph coloring, we use a polynomial reduction from another NP-complete problem known as the word problem for products of symmetric groups (WPPSG). |
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30.03.51748 Marcelo Campos University of Oxford |
Informatyka Teoretyczna An exponential improvement for diagonal Ramsey |
The Ramsey number R(k) is the minimum n such that every red-blue colouring of the edges of the complete graph Kn on n vertices contains a monochromatic copy of Kk. We prove that R(k)≤3.99k. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.
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11.09.35316 Bartłomiej Błoniarz |
Optymalizacja Kombinatoryczna (Some of) the many uses of Eulerian graphs in graph theory (plus some applications) |
The article showcases diverse associations between Eulerian graphs and other attributes of graphs such as being Hamiltonian, nowhere-zero flows, the cycle-plus-triangles problem, and issues emanating from it. It shows the application of compatible cycle decompositions in creating loopless 4-regular graphs with exactly one Hamiltonian cycle, or in establishing the equivalence between the Chinese Postman Problem and the Planar Bridgeless Minimum Cycle Covering Problem. |
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16.01.35266 Kacper Topolski, Jakub Wąs |
Simple and Faster Algorithms for Knapsack |
| Na tym seminarium zdefiniujemy problem plecakowy oraz jego wariacje - wersję 0-1, ograniczoną oraz DiffKnapsack. Przybliżymy najnowsze rezultaty związane z tym problemem. W szczególności zaprezentujemy prosty algorytm randomizowany rozwiązujący dyskretny wariant problemu plecakowego oraz oparty na nim algorytm rozwiązujący wersję ograniczoną. Jest on rozwinięciem pierwszego algorytmu o liniowej zależności względem liczby elementów, zaprezentowanego m.in. przez Adama Polaka. |
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16.01.35266 Kacper Topolski, Jakub Wąs |
Simple and Faster Algorithms for Knapsack |
| Na tym seminarium zdefiniujemy problem plecakowy oraz jego wariacje - wersję 0-1, ograniczoną oraz DiffKnapsack. Przybliżymy najnowsze rezultaty związane z tym problemem. W szczególności zaprezentujemy prosty algorytm randomizowany rozwiązujący dyskretny wariant problemu plecakowego oraz oparty na nim algorytm rozwiązujący wersję ograniczoną. Jest on rozwinięciem pierwszego algorytmu o liniowej zależności względem liczby elementów, zaprezentowanego m.in. przez Adama Polaka. |
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23.11.32582 Krzysztof Potępa Jagiellonian University |
Informatyka Teoretyczna Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs |
We develop a framework for algorithms finding diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) sub-quadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al., improving their technique. With our approach the algorithms become simpler and faster, working in Õ(k·V1-1/d·E) time complexity, where k is the diameter, d is the VC-dimension. Furthermore, it allows us to use the technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we answer a question posed by Bringmann et al., finding a Õ(n7/4) parameterized diameter algorithm for unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs. This is joint work with Lech Duraj and Filip Konieczny. |
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07.05.16151 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Some open problems from combinatorics and algorithmics |
The first presented problem concerns the majority coloring of graphs in the undirected and directed cases. A surprising connection with the problem of spreading epidemics in graphs will be shown. The second presented problem concerns the hat guessing game. The most classic results as well as the most interesting unresolved hypotheses will be shown. The last presented problem will concern randomized online algorithms for finding matching in bipartite graphs. Classic algorithms and research directions worth pursuing will be presented. |
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19.07.13417 Avi Wigderson Institute for Advanced Study, Princeton |
Informatyka Teoretyczna The Value of Errors in Proofs |
Recently, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", surprising and impacting not only complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the "Connes' embedding conjecture" in the area of von-Neumann algebras, and the "Tsirelson problem" in quantum information theory. It further connects Turing's seminal 1936 paper which defined algorithms to Einstein's 1935 paper with Podolsky and Rosen which challenged quantum mechanics. You can find the paper here https://arxiv.org/abs/2001.043 |
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28.05.43542 Julia Biały |
Optymalizacja Kombinatoryczna A game generalizing Hall's Theorem |
Authors investigate starting positions in a particular two-player game, considering scenarios where the first player can have a winning strategy. This work offers a broader interpretation of Hall's Theorem using Vizing's Theorem on edge-coloring in a specialized setting. |
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31.01.43530 Łukasz Selwa |
Optymalizacja Kombinatoryczna Orientations of Graphs with Prescribed Weighted Out-Degrees |
We study the complexity of the orientation problem where the out-neighborhood of every vertex is bounded by some function. This problem can be used to apply Galvin’s kernel method to show that graph G satisfies a certain coloring property. We show that the problem is NP-complete in the case of graphs that are bipartite, planar, and of maximum degree at most 3. We also prove some results on the (f,g)-choosability problem for weighted graphs, including a generalization of Brooks's theorem for weighted graphs. |
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13.04.40796 Fabrizio Frati Università Roma Tre |
Informatyka Teoretyczna Currents Trends and Hot Problems in Graph Drawing |
In this expository talk, I will discuss the topics that have attracted the most attention in the graph drawing community in recent years. The talk will try to convey the direction where the research in graph drawing is going, with a focus on the most intriguing open problems in the field. |
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07.12.21630 Michał Seweryn Université Libre de Bruxelles |
Informatyka Teoretyczna Recent results in graph product structure theory |
Graph product structure theory describes complicated graphs as subgraphs of strong products of simpler building blocks. Usually, the strong product involves three graphs: a graph of bounded treewidth, a small complete graph, and a path. For example, Dujmović et al. showed that every planar graph is a subgraph of the strong product of a treewidth-3 graph, a complete graph on three vertices, and a path. This theorem has been the key to solving many long-standing problems about planar graphs, and is arguably the most important result of the graph product structure theory. In my talk I will discuss some of the recent results in this field. First I will discuss two generalizations of the product structure theorem for planar graphs to k-planar graphs and k-powers of planar graphs with bounded degree. The distinguishing property of these theorems is that the bound on the treewidth in the product is an absolute constant independent of k and the maximum degree. Then, I will discuss some product structure theorems, where an n-vertex graph is a subgraph of the strong product of two graphs: a graph of constant treewidth, and a complete graph on O(√n) vertices. These theorems are qualitative strengthenings of √n-separator theorems by Lipton-Tarjan and Alon-Seymour-Thomas. Joint works with Marc Distel, Vida Dujmović, David Eppstein, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, and David Wood |
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13.08.87340 Ola Svensson École Polytechnique Fédérale de Lausanne |
Informatyka Teoretyczna The Price of Explainability for Clustering |
Given a set of points in d-dimensional space, an explainable clustering is one where the clusters are specified by a tree of axis-aligned threshold cuts. Dasgupta et al. (ICML 2020) posed the question of the price of explainability: the worst-case ratio between the cost of the best explainable clusterings to that of the best clusterings.
We show that the price of explainability for k-medians is at most 1+Hk−1; in fact, we show that the popular Random Thresholds algorithm has exactly this price of explainability, matching the known lower bound constructions. We complement our tight analysis of this particular algorithm by constructing instances where the price of explainability (using any algorithm) is at least (1−o(1))·ln k, showing that our result is best possible, up to lower-order terms. We also improve the price of explainability for the k-means problem to O(k·lnln k) from the previous O(k·ln k), considerably closing the gap to the lower bounds of Ω(k).
Finally, we study the algorithmic question of finding the best explainable clustering: We show that explainable k-medians and k-means cannot be approximated better than O(ln k), under standard complexity-theoretic conjectures. This essentially settles the approximability of explainable k-medians and leaves open the intriguing possibility to get significantly better approximation algorithms for k-means than its price of explainability.
This is joint work with Anupam Gupta, Madhusudhan Reddy Pittu, and Rachel Yuan |
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22.05.70921 Katarzyna Król |
Optymalizacja Kombinatoryczna Ball Packings and Lorentzian Discrete Geometry |
The problem of packing balls is to find an arrangement of spheres in space so that no spheres overlap. It is popular in the literature to consider packing disks - i.e. two-dimensional spheres. A tangency graph is a graph whose vertices are balls and whose edge is between vertices u and v if ball u and ball v touch each other. We study ball packings whose tangency graph is a higher dimensional grid graph. We give a loose bound on the size of such grid graphs that admit a ball packing. |
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25.01.70909 Jędrzej Kula |
Optymalizacja Kombinatoryczna Playing cards with Vizing's demon |
The paper's authors present a parametrized version of the solitaire game. In this version, players play against a demon whose task is to rearrange cards after each move in a way that the player will not be able to win the game. By defining a specific demon strategy and finding the winning strategy against it, one may prove König and Vizing theorems. |
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08.04.68175 Csaba Tóth California State University, Northridge |
Informatyka Teoretyczna Optimal spanners in Euclidean spaces |
For a set S of n points in a metric space (X,d) and a parameter t>1, a t-spanner is a weighted graph G=(S,E) such that the shortest path distance in G approximates the pairwise distances in the metric space up to a factor of at most t (stretch factor). This talk focuses on the d-dimensional Euclidean space in the regime where t is close to 1. Recent research uncovered tight trade-offs for two important quality measures for t-spanners: the sparsity |E(G)|/n and the lightness w(G)/w(MST(S)). We present an algorithm that constructs a t-spanner for a given set of n points in Euclidean d-space, by sparsifying classical Yao-graphs, that attains a worst-case optimal bound on the weight of the spanner. In the online model, a sequence of points arrive one-by-one, and we need to maintain a t-spanner for the first n points for all n. By combining sparse Yao-graphs and hierarchical clustering, we obtain an online algorithm that maintains a light spanner and achieves logarithmic competitive ratio compared to the offline optimum. |
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15.01.51756 Krzysztof Barański |
Optymalizacja Kombinatoryczna A note on degree-constrained subgraphs |
Last semester I presented a paper “A note on polynomials and f-factors of graphs” by Hamed Shirazi and Jacques Verstraëte, who proved two f-factor theorems using the Combinatorial Nullstellensatz. In this work, authors take a closer look at the same theorems and prove them in a different way. |
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20.09.51743 Filip Konieczny |
Optymalizacja Kombinatoryczna On constructive methods in the theory of colour-critical graphs |
k-critical graph is not (k-1)-colorable but every proper subgraph is. The authors take a constructive approach to the theory of critical graphs and show methods of how such graphs can be constructed, composed, and augmented, additionally discussing the advantages and drawbacks of these methods. |
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02.12.49009 John Iacono Université Libre de Bruxelles |
Informatyka Teoretyczna The pursuit of the dynamic optimality conjecture via the geometry of binary search trees |
In 1983, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence — any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any other tree algorithm is known to be dynamically optimal. In doing so we will present the geometric view of binary search trees, introduced in 2009, where we (with Erik D. Demaine, Dion Harmon, Daniel M. Kane and Mihai Pătraşcu) showed an equivalence between binary search tree algorithms and a simple combinatorial property of points in the plane. Almost all recent progress, which we will survey, towards the forty-year-old dynamic optimality conjecture since then has used this view, as it greatly simplifies reasoning about binary search trees. |
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09.09.32590 Rafał Pyzik |
Optymalizacja Kombinatoryczna Improved lower bounds on the number of edges in list critical and online list critical graphs |
A graph G is k-critical if it is not (k-1)-colorable, but every proper subgraph of G is. Authors improve the bound by Kostochka and Stiebitz for a number of edges in k-critical graphs. The same bound holds for k-list-critical and online k-list-critical graphs improving the bound established by Riasat and Schauz. This result follows from analyzing Alon-Tarsi orientable induced subgraphs satisfying certain conditions.
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15.05.32578 Aleksander Katan |
Optymalizacja Kombinatoryczna A not 3-choosable planar graph without 3-cycles |
An L-list coloring of graph G is a proper vertex coloring in which every vertex receives a color from a prescribed list L(v). G is said to be k-choosable, if all lists L(v) have cardinality k, and G is L-colorable for any assignment of those lists. The author presents a planar graph without 3-cycles that is not 3-choosable. We will also discuss other topics related to list colorings, such as the fact that every planar graph is 5-choosable.
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27.07.29844 Clément Rambaud École Normale Supérieure, PSL Paris |
Informatyka Teoretyczna Neighborhood complexity of planar graphs |
In a class of graphs of bounded expansion, for every graph in the class, for every non-empty set A of vertices, for every radius r, the number of distinct intersections between A and a ball of radius r is at most f(r)·|A| for some function f depending only on the considered class [Reidl, Sánchez Villaamil and Stravopoulos, 2019]. The function f coming from existing proofs is typically exponential. We prove that in the special case of planar graphs, f can be taken to be a polynomial, and more precisely in O(r4). We also show that a polynomial bound holds more generally for every proper minor-closed class of graphs. This is joint work with Gwenaël Joret. |
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05.05.13425 Rafał Kilar |
Optymalizacja Kombinatoryczna On the structure of k-connected graphs without K_k-minor |
The famous Hadwiger's Conjecture states that every k-chromatic graph must contain the clique Kk as a minor. It remains unproven for k>6. Motivated by this conjecture we can ask about the structure of k-connected graphs without Kk as a minor. We show that any such graph can't have three (k-2)-cliques that share at most three vertices, which is a generalization of previous results in the area. |
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08.01.13413 Bartłomiej Błoniarz |
Optymalizacja Kombinatoryczna Pólya's Permanent Problem |
The permanent of a square matrix is a function very similar to the determinant. It has important applications in counting problems, but computing it is a #P-complete problem. In 1913, Pólya proposed a method to calculate permanents using determinants, which was soon proven to be faulty in certain cases. This led to the question of when Pólya's method can be used, known as Pólya's Permanent Problem. The article provides an overview of the problem, including equivalent versions and a solution to one of the formulations. |
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13.02.76397 Demian Banakh |
Optymalizacja Kombinatoryczna Flip distance to a non-crossing perfect matching |
A non-crossing perfect matching is Euclidean matching on 2n points so that no 2 segments cross. Given some crossing matching, we can iteratively apply the flip operation (fix any 2 crossing segments, and swap the endpoints so that they don't cross) to eventually arrive at a non-crossing matching. We will investigate the upper and lower bounds for the number of flips sufficient and necessary to eliminate all crossings. It is conjectured that θ(n2) flips are always sufficient.
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19.10.76384 Szymon Salabura |
Optymalizacja Kombinatoryczna Edge lower bounds for list critical graphs, via discharging |
We say that a graph G is k-choosable if G has a proper coloring from every list assignment L with |L(v)|=k for every vertex v. A graph G is k-list-critical if it's not k-choosable, but every proper subgraph of G is. The problem of bounding the number of edges from below in such graphs has been widely studied, starting with the work of Gallai. The authors present a rephrased version of his proof using the discharging method and improve the original result by presenting additional properties of such graphs, enabling a different set of discharging rules. |
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24.02.76334 Justyna Jaworska, Jakub Siuta |
Simple, deterministic, fast (but weak) approximations to edit distance and Dyck edit distance |
| Dla problemów znjadowania odległości edycyjnej i odległości edycyjnej Dycka chcemy znaleść szybkie, deterministyczne i proste aproksymacje, z być może dużym współczynnikiem aproksymacji. Dla klasycznej odległości edycyjnej wprowadzimy klasę szybkich i prostych algorytmów nazywanych "algorytmami pojdedycznego skanowania". Saha, w 2014. roku, podał randomizowany algorytm z tej klasy osiągający aproksymację O(d) dla słow x, y takich że ich ogległość edycyjna jest rzędu O(d). W tej pracy prezentujemy: (1) deterministyczny algorytm z wymienionej klasy osiągający podobne rezultaty oraz (2) dowodzimy, że nie istnieje (nawet randomizowany) algorytm z tej klasy, który dawałby lepszą aproksymację. Dla odległości edycyjnej Dycka, Saha zaproponował randomizowaną redukcję z odległości edycyjnej Dycka do klasycznej odległości edycyjnej o koszcie O(log d), gdzie d to odległość edycyjna Dycka. Podamy redukcję deterministyczną której zarówno sfromułowanie jak i udowodnienie poprawności jest prostsze. |
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31.12.73650 David Eppstein University of California, Irvine |
Informatyka Teoretyczna The Complexity of Iterated Reversible Computation |
Reversible computation has been studied for over 60 years as a way to evade fundamental physical limits on the power needed for irreversible computational steps, and because quantum computing circuits are necessarily reversible. We study a class of problems based on computing the iterated values of a reversible function. The story leads through Thomason's lollipop algorithm in graph theory, circuit complexity, and reversible cellular automata, to card shuffling, the reflections of light in jewels, and curves on topological surfaces, and involves both PSPACE-hard problems and problems with unexpected polynomial-time algorithms. |
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09.10.57231 Piotr Kaliciak |
Optymalizacja Kombinatoryczna Decomposing 4-connected planar triangulations into two trees and one path |
A graph is 4-connected if no matter which 4 vertices we remove from it, it remains connected. We can decompose every 4-connected planar triangulation into a Hamiltonian path and two trees. Moreover, we can decompose any Hamiltonian planar triangulation into two trees and one spanning tree of degree at most 3. These results are best-possible, this means that we cannot decrease the maximum degree of the first tree. |
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14.06.57219 Kamil Galewski |
Optymalizacja Kombinatoryczna On the discrepancy of circular sequences of reals |
The discrepancy is a function that measures the irregularity of the distribution of a given sequence of real numbers. The authors present a new method to measure discrepancy for sequences of reals lying on a circle of circumference 1, as a more sensitive alternative to the previously known functions. They also show a tight upper bound for this function. |
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25.08.54485 Pat Morin Carleton University |
Informatyka Teoretyczna Proof of the Clustered Hadwiger Conjecture |
Hadwiger's Conjecture asserts that every Kh-minor-free graph is properly (h-1)-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed h, every Kh-minor-free graph is (h-1)-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed t≥s, we show that every Ks,t-minor-free graph is (s+1)-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. This joint work with Vida Dujmović, Louis Esperet, and David R. Wood. |
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03.06.38066 Łukasz Gniecki |
Optymalizacja Kombinatoryczna Sequences of points on a circle |
Consider a sequence of points a1, a2, a3, ... on a circle with radius 1/(2π), in other words, numbers mod 1. The numbers a1, a2, ..., an define n intervals with a total length of 1. Denote by M[n,r](a) and m[n,r](a) the largest and the smallest length of consecutive r intervals. We can think of how the values n·M[n, r](a) and n·m[n, r](a) will behave if we go with n to infinity. In particular, for a given sequence a we can find the upper limit of n·M: L[r](a) = limsup n·M[n,r](a) and the lower limit of n·m: l[r](a) = liminf n·m[n,r](a). We can go further and consider the greatest lower bound on L[r](a) (g.l.b. in short) and the lowest upper bound on l[r](a) (l.u.b. in short), overall sequences a. The authors derive bounds on this g.l.b. and l.u.b. and in the case of r = 1, they prove these bounds are tight by giving an example of a sequence a which satisfies these bounds. |
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06.02.38054 Ignacy Buczek |
Optymalizacja Kombinatoryczna 10 Problems for Partitions of Triangle-free Graphs |
The original sparse halves conjecture of Erdos, formed in 1976, states that every triangle-free graph has a subset of n/2 vertices with at most n2/50 edges. As it still remains unsolved, a number of related problems have been stated in order to better understand the problems of partitioning graphs into sparse subsets. In our work, we present and improve the results of some of the existing problems of this kind, and in addition, we state multiple new ones and provide initial results. |
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14.06.38003 Rafał Pyzik, Sebastian Spyrzewski |
Treewidth is NP-Complete on Cubic Graphs (and related results) |
| Autorzy pracy podają prosty dowód NP-zupełności problemu Treewidth, udowadniając jego NP-zupełność w klasie dopełnień grafów dwudzielnych. Praca poprawia też rezultat Bodlaedera i Thilikosa z roku 1997 mówiący, że Treewidth jest NP-zupełne w grafach o maksymalnym stopniu co najwyżej 9, pokazując NP-zupełność w grafach regularnych stopniu 3. |
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20.04.35320 Ruta Mehta University of Illinois at Urbana-Champaign |
Informatyka Teoretyczna Competitive division of goods, bads, and mixed: existence, computation, and complexity |
Fair division is the problem of allocating a set of items among agents in a fair and efficient manner. This age-old problem, mentioned even in the Bible, arises naturally in a wide range of real-life settings, for example, school seat assignments, partnership dissolution, sharing of satellites, and dividing costs for climate resilience. Division based on competitive equilibrium (CE) has emerged as one of the best mechanisms for this problem. The existence and computability of CE have been extensively studied when all items are disposable goods, while the problem is less explored when some of them are non-disposable chores (bads). In this talk, I will discuss recent algorithmic advances on the computation of CE when each item may be a good, a chore, or both (mixed). I will first consider the case of additive valuations, where when all items are goods, the CE set is well-known to be captured by convex programming formulations and thereby forms a convex set. In sharp contrast, with chores, the CE set may be nonconvex and disconnected. I will discuss how to handle this non-convexity through a new exterior-point method to find an approximate CE in polynomial time (FPTAS). This method seems general enough to work with any mathematical formulation that optimizes a coordinate-wise monotone function over linear constraints. Finally, I will discuss recent developments on the exchange setting (barter system) on existence and computational complexity. Based on joint works with Shant Boodaghians, Bhaskar Ray Chaudhury, Jugal Garg, and Peter McGlaughlin. |
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14.12.16154 Ralph Keusch Siemens Mobility CH |
Informatyka Teoretyczna A Solution to the 1-2-3 Conjecture |
In 2004, Karoński, Łuczak and Thomason conjectured that for each connected graph on at least 3 vertices, it is possible to assign weights from {1,2,3} to the edges such that neighboring vertices always obtain different weighted degrees. Recently, Przybyło verified the conjecture for all graphs G where the minimum degree is sufficiently large, compared to the maximum degree and to an absolute constant. In general, the best-known bound was by Kalkowski, Karoński, and Pfender from 2011. They proved that such an assignment is always possible with the weight set {1,2,3,4,5}. We present a flow-based strategy to construct vertex-coloring edge-weightings and show how it was first used to shrink the general bound to the set {1,2,3,4} and now led to the confirmation of the conjecture. |
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04.10.84610 Jakub Siuta |
Optymalizacja Kombinatoryczna List-avoiding orientations |
Given a graph G with a set F(v) of forbidden values at each v∈V(G), an F-avoiding orientation of G is an orientation in which deg+(v)∉F(v) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if |F(v)|<12deg(v) for each v∈V(G), then G has an F-avoiding orientation, and they showed that this statement is true when 12 is replaced by 14. In this paper, we take a step toward this conjecture by proving that if |F(v)|<⌊13deg(v)⌋ for each vertex v, then G has an F-avoiding orientation. Furthermore, we show that if the maximum degree of G is subexponential in terms of the minimum degree, then this coefficient of 13 can be increased to 21/2−1−o(1) ≈ 0.414. Our main tool is a new sufficient condition for the existence of an F-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi. |
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08.06.84598 Grzegorz Gawryał |
Optymalizacja Kombinatoryczna Critically paintable, choosable or colorable graphs |
The concept of criticality in graphs was introduced around 1950 to capture the essence of a graph that is not colorable with k colors. Since then, the idea became more and more popular in the literature. We will generalize the results about criticality to list and online list coloring of graphs, using a stronger version of Brooks and Gallai's theorems and prove their implications for graphs drawn on different surfaces, basically showing, that for any surface and k ≥ 5, there are always only finitely many critical graphs for both paintability and choosability. |
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14.10.84547 Łukasz Grobelczyk, Rafał Loska |
This Game is Not Going to Analyze Itself |
| Praca analizuje kilka problemów powiązanych z grą przeglądarkową "This Game Is Not Going To Load Itself", w której gracz ma za zadanie przekierować poruszające się kolorowe kwadraty do odpowiedniego ujścia na planszy poprzez ustawianie na niej kolorowych strzałek. Problem decyzyjny czy można wygrać grę jest w klasie $\Sigma_2^P$, jest NP-trudny w wersji offline, a nawet bez możliwości układania strzałek przez gracza jest zarówno NP- jak i coNP-trudny. Praca analizuje również problem istnienia strategii wygrywającej. |
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20.08.81864 Alex Scott University of Oxford |
Informatyka Teoretyczna On a problem of El-Zahar and Erdős |
Two subgraphs A,B of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results. We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number. Second, we look at what happens if we replace the hypothesis that G has large chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding Kt, is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding a complete graph we exclude a complete bipartite graph. Finally, we discuss analogous problems for tournaments. This is joint work with Tung Nguyen and Paul Seymour. |
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29.05.65445 Tomasz Mazur |
Optymalizacja Kombinatoryczna A note on large induced subgraphs with prescribed residues in bipartite graphs |
A known result of Scott is that for every k ≥ 2, there exists a constant c(k) > 0 such that every bipartite n-vertex graph G without isolated vertices has an induced subgraph H on at least c(k)·n vertices such that degH(v) = 1 (mod k) for every vertex v in H. Scott conjectured that c(k) = Ω(1/k). A confirmation of this conjecture is supplied in this paper. |
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01.02.65433 Katzper Michno |
Optymalizacja Kombinatoryczna Dimension and cut vertices: an application of Ramsey theory |
Dimension of a poset P (denoted dim(P)), is the smallest natural number d, such that there are d linear extensions of P s.t. their intersection is exactly P. Among many results regarding the poset dimension, there are quite a few that find relationships between the dimension and some properties of its cover graph. We will discuss one such result, that if for every block B in the cover graph of P, the induced subposet of P with ground set B has dimension at most d, then dim(P) ≤ d+2. We will also show constructions of examples proving that this bound is tight using Product Ramsey Theorem. |
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15.04.62699 Martin Grohe RWTH Aachen |
Informatyka Teoretyczna A Deep Dive into the Weisfeiler-Leman Algorithm |
The Weisfeiler-Leman algorithm is a well-known combinatorial graph isomorphism test going back to work of Weisfeiler and Leman in the late 1960s. The algorithm has a surprising number of seemingly unrelated characterisations in terms of logic, algebra, linear and semi-definite programming, and graph homomorphisms. Due to its simplicity and efficiency, it is an important subroutine of all modern graph isomorphism tools. In recent years, further applications in linear optimisation, probabilistic inference, and machine learning have surfaced. In my talk, I will introduce the Weisfeiler-Leman algorithm and some extensions. I will discuss its expressiveness and the various characterisations, and I will speak about its applications. |
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27.09.46267 Jakub Dziarkowski |
Optymalizacja Kombinatoryczna Research problems |
We will discuss selected open problems in discrete mathematics. Two of them are connected to discrete geometry: Piercing families of planar convex sets - finding the minimum number of points needed to pierce a collection of convex sets in the plane, Splitting lines for planar point sets - splitting set of points equally by line through points of this set. Other are graph theory problems: Acyclic edge-coloring of graphs, Two questions on long cycles, and Representations of graphs modulo n. |
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09.12.43533 Sebastian Siebertz Universität Bremen |
Informatyka Teoretyczna Advances in algorithmic meta-theorems |
Algorithmic meta-theorems provide general explanations when and why certain algorithmic problems can be solved efficiently. They are typically formulated in terms of logic (defining the descriptive complexity of the problems) and structural properties of their inputs. A prototypical algorithmic meta-theorem is Courcelle’s Theorem, stating that every graph property definable in monadic second-order logic (MSO) can be decided in linear time on every graph class of bounded treewidth. Similarly, every graph property definable in first-order logic (FO) can be tested efficiently on every nowhere dense graph class. In this talk I will present recent progress on algorithmic meta-theorems for FO on dense graph classes as well as for logics whose expressive power lies between MSO and FO. The presented results reveal beautiful connections between structural graph theory, classical model theory and algorithmics. |
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23.05.27102 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Obstacle Number of Graphs |
An obstacle is a connected shape on the plane. Given a set of obstacles and a set of points on the plane, we can define a visibility graph on the set of points. Two points are connected by an edge if a straight line between them is disjoint from all the obstacles. We say that the set of points and obstacles is an obstacle representation of the resulting graph. We define the obstacle number of a graph as the minimum number of obstacles needed to represent the graph in an obstacle representation. This parameter was introduced by Alpert, Koch, and Laison in 2011. I will discuss many examples of graphs and their obstacle numbers. I will also present a related notion of convex obstacle number. Moreover, during the presentation, I will state many interesting open problems. |
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03.08.24368 Sándor Kisfaludi-Bak Aalto University |
Informatyka Teoretyczna On geometric variants of TSP and Steiner tree |
In the classic Euclidean traveling salesman problem, we are given n points in the Euclidean plane, and the goal is to find the shortest round trip that visits all the points. We will discuss some of the key techniques that allowed us to find (conditionally) optimal exact and approximation algorithms for this problem, while the closely related Steiner tree problem seems to resist many similar attempts. We will then turn to the traveling salesman or Steiner tree with "neighborhoods". Here instead of points, we are given a set of affine subspaces, and the goal is to find the shortest round trip or tree that intersects each subspace. It turns out that these problems have a different computational complexity than the classic problems with points: they require a completely novel approach for the hyperplane case, while the other cases remain largely unresolved. |
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29.03.5203 Mikkel Thorup University of Copenhagen |
Informatyka Teoretyczna Reconstructing the Tree of Life (Fitting Distances by Tree Metrics) |
We consider the numerical taxonomy problem of fitting an SxS distance matrix D with a tree metric T. Here T is a weighted tree spanning S where the path lengths in T induce a metric on S. If there is a tree metric matching D exactly, then it is easily found. If there is no exact match, then for some k, we want to minimize the Lk norm of the errors, that is, pick T so as to minimize ||D-T||k = (Σi,jϵS |D(i,j)-T(i,j)|k) 1/k. An evolutionary tree induces a hierarchical classification of species and this is not just tied to biology. Medicine, ecology and linguistics are just some of the fields where this concept appears, and it is even an integral part of machine learning and data science. Fundamentally, if we can approximate distances with a tree, then they are much easier to reason about: many questions that are NP-hard for general metrics can be answered in linear time on tree metrics. In fact, humans have appreciated hierarchical classifications at least since Plato and Aristotle (350 BC). The numerical taxonomy problem is important in practice and many heuristics have been proposed. In this talk we will review the basic algorithmic theory, results and techniques, for the problem, including the most recent result from FOCS'21 [Vincent Cohen-Addad et al., 2021]. They paint a varied landscape with big differences between different moments, and with some very nice open problems remaining. - At STOC'93, Farach, Kannan, and Warnow [Martin Farach et al., 1995] proved that under L∞, we can find the optimal ultrametric. Almost all other variants of the problem are APX-hard - At SODA'96, Agarwala, Bafna, Farach, Paterson, and Thorup [Richa Agarwala et al., 1999] showed that for any norm Lk, k≥1, if the best ultrametric can be α-approximated, then the best tree metric can be 3α-approximated. In particular, this implied a 3-approximation for tree metrics under L∞. - At FOCS'05, Ailon and Charikar [Nir Ailon and Moses Charikar, 2011] showed that for any Lk, k≥1, we can get an approximation factor of O(((log n)(log log n))1/k) for both tree and ultrametrics. Their paper was focused on the L1 norm, and they wrote "Determining whether an O(1) approximation can be obtained is a fascinating question". - At FOCS'21, Cohen-Addad, Das, Kipouridis, Parotsidis, and Thorup [Vincent Cohen-Addad et al., 2021] showed that indeed a constant factor is possible for L1 for both tree and ultrametrics. This uses the special structure of L1 in relation to hierarchies. - The status of Lk is wide open for 1<k<∞. All we know is that the approximation factor is between Ω(1) and O((log n)(log log n)). |
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19.12.73658 Krzysztof Barański |
Optymalizacja Kombinatoryczna A note on polynomials and f-factors of graphs |
A k-factor of a graph is a spanning k-regular subgraph. Here, we will focus on a more general term: f-factors of graphs, where f is a function assigning to each vertex of the graph a set of integers from 0 to the degree of that vertex, and f-factor is a spanning subgraph of the graph, where for every vertex v, degree of v is an element of f(v). Authors show a necessary condition for such f-factors. |
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24.08.73646 Demian Banakh |
Optymalizacja Kombinatoryczna Token sliding on graphs of girth five |
In the Token sliding problem, one starts with a graph and independent sets Is, It. We put k tokens on vertices of Is and ask whether it's possible to reach It after a finite sequence of moves, where 1 move is sliding 1 token along the edge so that no 2 tokens are adjacent at any point. It was shown in 2021 that this problem is W[1]-hard for graphs of girth 4 or less. In this presentation, we will see how the problem becomes Fixed-parameter tractable for the other graphs (girth 5 or more). |
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30.12.73595 Grzegorz Gawryał, Szymon Salabura |
TSP in a Simple Polygon |
| Problem komiwojażera (TSP) jest jednym z najbardziej popularnych problemów optymalizacyjnych w algorytmice. Jest on NP-trudny, nawet wtedy, gdy graf na wejściu jest grafem odległości euklidesowych między danymi punktami na płaszczyźnie. Autorzy wprowadzają nowy wariant tego problemu - TSP w wielokącie prostym, w którym to problemie należy znaleźć najkrótszą trasę nie wychodzącą poza wielokąt i odwiedzającą pewien zbiór punktów w tym wielokącie, w dowolnej kolejności. Autorzy najpierw pokazują, jak zastosować ogólniejszy i dość skomplikowany algorytm Marxa, Pilipczuka i Pilipczuka do tego problemu, uzyskując złożoność poly(n,m) + 2(O(sqrt(n) log n)), a następnie prezentują własny, znacznie prostszy algorytm rozwiązujący ten wariant TSP w tej samej złożoności. |
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05.11.70912 Andrzej Grzesik Jagiellonian |
Informatyka Teoretyczna Turán-type problems for directed cycles |
A standard Turán problem for a graph F is to determine the maximal number of edges in a graph not containing F as a subgraph. This problem for directed cycles in oriented graphs is trivial, but its various generalizations, when one asks for minimum outdegree or number of other subgraphs, occurred to be hard problems. In particular, finding minimum outdegree (or semidegree) forcing an oriented graph to contain a directed triangle is a Caccetta-Häggkvist conjecture, which is open for 45 years despite numerous partial results. During the talk we will present a solution (obtained with Jan Volec) to a conjecture of Kelly, Kühn and Osthus on the minimum semidegree forcing an oriented graph to contain a directed cycle of any given length at least four. We will also discuss results (obtained jointly with Justyna Jaworska, Bartłomiej Kielak and Tomasz Ślusarczyk) for the generalized Turán problem for directed cycles when one maximizes the number of directed cycles of some other length. |
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21.03.70803 Krzysztof Barański |
Podstawy Informatyki A verified framework for higher-order uncurrying optimizations by Zaynah Dargaye and Xavier Leroy |
| Function uncurrying is an important optimization for the efficient execution of functional programming languages. This optimization replaces curried functions by uncurried, multiple-argument functions, while preserving the ability to evaluate partial applications. First-order uncurrying (where curried functions are optimized only in the static scopes of their definitions) is well understood and implemented by many compilers, but its extension to higher-order functions (where uncurrying can also be performed on parameters and results of higher-order functions) is challenging. This article develops a generic framework that expresses higher-order uncurrying optimizations as type-directed insertion of coercions, and prove its correctness. The proof uses step-indexed logical relations and was entirely mechanized using the Coq proof assistant. |
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13.08.54493 Bartłomiej Błoniarz |
Optymalizacja Kombinatoryczna A Survey of the Game “Lights Out!" |
In the most common version of the Lights Out problem, we have an undirected graph G, in which every vertex represents a light either on or off. We can toggle any light, but such action is always followed by all the neighboring lights also switching. The goal is to decide whether it is possible to turn all the lights off. The authors collected many results regarding this problem to present them in a unified framework. For example, they show proof that for any graph with all the lights initially on, it is possible to turn them off. They also study the optimization variants of the problem, such as finding the minimum number of lights we need to toggle, which they show to be NP-hard.
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18.04.54481 Jakub Dziarkowski |
Optymalizacja Kombinatoryczna Note on Perfect Forests |
A spanning forest F of a graph G is called a perfect forest if all its components are induced subgraphs of G and the degree of each vertex x in F is odd. It is easy to see that if a connected graph G has a perfect forest, then G is of even order. Interestingly, the opposite implication is also true (A.D. Scott was the first to prove it). Gregory Gutin gave a short proof of this theorem using linear algebra. |
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24.08.54430 Bartłomiej Wacławik, Krzysztof Ziobro |
Tiny Pointers |
| Praca wprowadza nowe pojęcie: wskaźniczek. Wskaźniczek jest obiektem, który pozwala na dostęp do jednego z n miejsc w pamięci, jednocześnie używając znacząco mniej niż log(n) bitów. Jest to możliwe dzięki użyciu wprowadzonej w pracy tablicy dereferencyjnej, która pozwala dla danego klucza k (z dużym prawdopodobieństwem) zaalokować komórkę pamięci zwracając wskaźniczek, który razem z kluczem pozwalaja uzyskać dostęp do zaalokowanej komórki w czasie stałym. Dodatkowo autorzy podają przykłady zastosowań w popularnych strukturach danych, których rozmiar można zredukować dzięki zastąpieniu klasycznych wskaźników wskaźniczkami. Wśród tych przykładów znajdują się między innymi drzewa BST oraz słowniki o stałej pojemności. |
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01.07.51747 Tuukka Korhonen University of Bergen |
Informatyka Teoretyczna An improved parameterized algorithm for treewidth |
Treewidth is a fundamental graph parameter that, informally, characterizes how tree-like a graph is. We give a 2O(k^2)·nO(1) time algorithm for determining if the treewidth of a given n-vertex graph is at most k and outputting the corresponding tree decomposition. This resolves the long-standing open problem of whether there is a 2o(k^3)·nO(1) time algorithm for treewidth. In particular, this is the first improvement on the dependency on k in fixed-parameter algorithms for treewidth since the 2O(k^3)·nO(1) time algorithm given in 1991 by Bodlaender and Kloks, and independently, by Lagergren and Arnborg. We also give a kO(k/ε)·nO(1) time (1+ε)-approximation algorithm for treewidth. Joint work with Daniel Lokshtanov. |
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13.11.51637 Roman Madej |
Podstawy Informatyki Modular Construction of Fixed Point Combinators and Clocked Bohm Trees by Jorg Endrullis, Dimitri Hendriks and Jan Willem Klop |
Fixed point combinators (and their generalization: looping combinators) are classic notions belonging to the heart of λ-calculus and logic. We start with an exploration of the structure of fixed point combinators (fpc’s), vastly generalizing the well-known fact that if Y is an fpc, Y (SI) is again an fpc, generating the B ̈ohm sequence of fpc’s. Using the infinitary λ-calculus we devise infinitely many other generation schemes for fpc’s. In this way we find schemes and building blocks to construct new fpc’s in a modular way. Having created a plethora of new fixed point combinators, the task is to prove that they are indeed new. That is, we have to prove their β-inconvertibility. Known techniques via B ̈ohm Trees do not apply, because all fpc’s have the same Bohm Tree (BT). Therefore, we employ ‘clocked BT’s’, with annotations that convey information of the tempo in which the data in the BT are produced. BT’s are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for λ-terms. The corresponding equality is strictly intermediate between =β and =BT, the equality in the classical models of λ-calculus. An analogous approach pertains to L ́evy–Longo and Berarducci trees. Finally, we increase the discrimination power by a precision of the clock notion that we call ‘atomic clock’.
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08.04.35328 Julia Biały |
Optymalizacja Kombinatoryczna Can a party represent its constituency? |
The paper focuses on the representation problem in political elections, using a theorem from number theory. A. Katz's work gives an answer to the question - of whether there exists a way to construct the election list so that it does not matter how many politicians are selected and the politically different groups of the party will be represented? |
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13.12.35315 Katzper Michno |
Optymalizacja Kombinatoryczna Internal Partitions of Regular Graphs |
We consider internal partitions of graphs, which is a partition of V into two sets, such that every vertex has at least half of its neighbors in its own set. Several investigators have raised the conjecture that d-regular graphs always have an internal partition, assuming their set of vertices is big enough. Here we prove this conjecture for d=6. We also investigate the case when |V|=d+4, which leads to some new problems on cubic graphs, and find new families of graphs that don't have an internal partition. |
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18.04.35265 Ignacy Buczek, Tomasz Buczyński |
List Colouring Trees in Logarithmic Space |
| Dla danego n-wierzchołkowego grafu G = (V, E) oraz listy L(v) ⊆ {1, ..., n} dozwolonych kolorów dla każdego wierzchołka v ∊ V, kolorowanie listowe jest kolorowaniem wierzchołkowym c grafu G spełniającym c(v) ∊ L(v) dla każdego v. Autorzy pracy dowodzą, że problem kolorowania listowego n-wierzchołkowych drzew może być rozwiązany za pomocą deterministycznej maszyny Turinga używającej O(log n) bitów na taśmie roboczej. |
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23.02.32582 Jonathan Narboni Jagiellonian |
Informatyka Teoretyczna Vizing's Conjecture Holds |
In 1964 Vizing proved that to properly color the edges of a graph G, one need at most ∆+1 colors, where ∆ is the maximum degree of G. In his paper, Vizing actually proves that one can transform any proper edge coloring into a (∆+1)-edge-coloring using only Kempe changes. Soon after his paper, he asked the following question: is an optimal edge-coloring always reachable from any proper edge-coloring using only Kempe changes? Bonamy & al. proved that the conjecture holds for triangle free graphs, following their work, we prove that it holds for all graphs. |
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08.07.32472 Rafał Loska |
Podstawy Informatyki Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study by Olivier Bodini, Antoine Genitrini, Cécile Mailler and Mehdi Naima |
| In this paper we study two models of labelled random trees that generalise the original unlabelled Schroder tree. Our new models can be seen as models for phylogenetic trees in which nodes represent species and labels encode the order of appearance of these species, and thus the chronology of evolution. One important feature of our trees is that they can be generated efficiently thanks to a dynamical, recursive construction. Our first model is an increasing tree in the classical sense (labels increase along each branch of the tree and each label appears only once). To better model phylogenetic trees, we relax the rules of labelling by allowing repetitions in the second model. For each of the two models, we provide asymptotic theorems for different characteristics of the tree (e.g. degree of the root, degree distribution, height, etc), thus giving extensive information about the typical shapes of these trees. We also provide efficient algorithms to generate large trees efficiently in the two models. The proofs are based on a combination of analytic combinatorics, probabilistic methods, and bijective methods (we exhibit bijections between our models and well-known models of the literature such as permutations and Stirling numbers of both kinds). It turns out that even though our models are labelled, they can be specified simply in the world of ordinary generating functions. However, the resulting generating functions will be formal. Then, by applying Borel transforms the models will be amenable to techniques of analytic combinatorics. |
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02.12.16162 Jakub Siuta |
Optymalizacja Kombinatoryczna On Induced Subgraphs with All Degrees Odd |
Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove that every connected graph of even order has a vertex partition into sets inducing subgraphs with all degrees odd, and give bounds for the number of sets of this type required for vertex partitions and vertex covers. We also give results on the partitioning and covering problems for random graphs. |
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07.08.16150 Aleksander Katan |
Optymalizacja Kombinatoryczna A generalization of Konig's theorem |
König's theorem lets us determine the maximum number of pairwise independent edges in a bipartite graph. In the paper, L. Lovász focuses on critical graphs, meaning that if any of their edges are removed, the size of maximum matching diminishes. Considering a certain generalization of the above-mentioned concept, Lovász gives a simple condition that is necessary and sufficient for a graph to be critical. The result is used to solve a conjecture by Erdős regarding strict hypergraph coloring. |
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19.10.13416 Michał Pilipczuk University of Warsaw |
Informatyka Teoretyczna Flipper games for monadically stable classes of graphs |
We will provide a gentle introduction to the on-going work on constructing a structural theory for graph classes defined by forbidding obstructions definable in logic. The focus will be on monadically stable classes of graphs: classes where one cannot define arbitrary long total orders using a fixed first-order formula. We will review recent advances on characterizing these classes in a purely combinatorial manner, in particular through a game model: the Flipper game. |
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04.03.13307 Sebastain Spyrzewski |
Podstawy Informatyki A characterization of lambda-terms transforming numerals by PAWEŁ PARYS |
| It is well known that simply typed λ-terms can be used to represent numbers, as well as some other data types. We show that λ-terms of each fixed (but possibly very complicated) type can be described by a finite piece of information (a set of appropriately defined intersection types) and by a vector of natural numbers. On the one hand, the description is compositional: having only the finite piece of information for two closed λ-terms M and N, we can determine its counterpart for M N, and a linear transformation that applied to the vectors of numbers for M and N gives us the vector for M N. On the other hand, when a λ-term represents a natural number, then this number is approximated by a number in the vector corresponding to this λ-term. As a consequence, we prove that in a λ-term of a fixed type, we can store only a fixed number of natural numbers, in such a way that they can be extracted using λ-terms. More precisely, while representing k numbers in a closed λ-term of some type, we only require that there are k closed λ-terms M1, . . . , M k such that M i takes as argument the λ-term representing the k-tuple, and returns the i-th number in the tuple (we do not require that, using λ-calculus, one can construct the representation of the k-tuple out of the k numbers in the tuple). Moreover, the same result holds when we allow that the numbers can be extracted approximately, up to some error (even when we only want to know whether a set is bounded or not). All the results remain true when we allow the Y combinator (recursion) in our λ-terms, as well as uninterpreted constants. |
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04.04.62707 Ignacy Buczek |
Optymalizacja Kombinatoryczna K4-free graphs have sparse halves |
In the extremal graph theory, there are many unsolved problems related to the finding of sparse subsets in graphs. The most famous one, stated by Erdos in 1976, asks whether every triangle-free graph contains n/2 vertices that span at most 1/50 n2 edges. In our work we consider, and successfully prove, a modified version of this theorem which conjectures that every K4-free graph has n/2 vertices spanning at most 1/18 n2 edges. This bound is tight, as the balanced blow-up of a triangle is an extreme example. We achieve the proof by strengthening some of the previous results and by stating some new arguments which show that the only K4-free graph which has at least 1/18 n2 edges in every half is the blow-up of a triangle. |
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07.12.62694 Łukasz Selwa |
Optymalizacja Kombinatoryczna Isomorphic bisections of cubic graphs |
Ando conjecture states that we can partition vertices of any cubic graph into two parts that induce isomorphic subgraphs. We show that this conjecture is true for sufficiently large connected cubic graphs. In the proof, we use probabilistic methods with recoloring arguments. |
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18.02.59961 Ross Kang University of Amsterdam |
Informatyka Teoretyczna Colouring graphs with sparse neighbourhoods |
Let us say that a graph of maximum degree Δ has local density at most η if the number of edges spanning any neighbourhood is at most η·(Δ choose 2), i.e. if the edge density is no more than an η fraction of the maximum possible. What is the largest chromatic number of such graphs? When η=0, this corresponds to asking about the largest chromatic number in triangle-free graphs of maximum degree Δ. This goes back to an old question of Vizing and is the objective of a recent breakthrough of Molloy. It is natural — and also connects to various other problems in the field — to consider other choices for η. We will broadly discuss this problem, including its classic origins in Ramsey theory, and some different ideas that have recently proven fruitful. This will touch on recent joint works with Davies, Hurley, de Joannis de Verclos, Pirot, and Sereni.
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04.07.59851 Łukasz Grobelczyk - canceled |
Podstawy Informatyki Bijections between planar maps and planar linear normal \lambda-terms with connectivity condition by Wenjie Fang |
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27.11.43541 Hubert Zięba |
Optymalizacja Kombinatoryczna The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture |
The 1-2-3 conjecture asserts that for every connected simple graph of order at least 3 edges can be weighted with 1,2 and 3 so that each pair of adjacent has different weighted degrees. We consider a modified version of this conjecture with 1,2 weights only. By using f-factors modulo k of the graph, we prove it for non-bipartite (6𝛘(G)-5)-edge-connected graphs and completely characterize bipartite graphs having this property. |
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02.08.43529 Tomasz Mazur |
Optymalizacja Kombinatoryczna Improved lower bound for the list chromatic number of graphs with no Kt minor |
Hadwiger's conjecture is an important conjecture in graph theory which states that every graph without a Kt-minor is (t-1)-colorable. This conjecture does not extend to list colorings, but Kawarabayashi and Mohar (2007) conjectured that there exists a constant c such that every graph with no Kt-minor has a list chromatic number at most c·t. More specifically, they conjectured that c = 3/2 is sufficient. Refuting the latter conjecture, we prove using the probabilistic method that there exist graphs with no Kt-minor with list chromatic number at least (2-o(1))·t, and hence c ≥ 2 is necessary. This improves the previous best-known lower bound by Barát, Joret, and Wood (2011), who proved that c ≥ 4/3. |
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07.12.43478 Roman Madej, Paweł Nowak |
Sinkless Orientation Made Simple |
| Sinkless Orientation jest problemem grafowym, polegającym na skierowaniu krawędzi w grafie, aby każdy wierzchołek o stopniu co najmniej trzy miał krawędź wychodzącą. Problem ten odgrywa kluczową rolę w zrozumieniu teorii obliczeń rozproszonych. Tematem rozważań pracy będzie analiza lokalności problemu, jednej z podstawowej własności rozproszonych algorytmów grafowych, w modelach LOCAL i SLOCAL. Znane jest już dokładne ograniczanie w modelu LOCAL oraz ograniczenie górne w modelu SLOCAL, natomiast standardowe dowody wykorzystują zaawansowane techniki. W pracy autorzy prezentują jednak nowe, elementarne i samowystarczalne dowody obydwu ograniczeń. |
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14.10.40795 Boris Bukh Carnegie Mellon |
Informatyka Teoretyczna Extremal graphs without exponentially-small bicliques |
In 1954 Kővári, Sós, and Turán showed that every n-vertex graph not containing Ks,t has at most O(n2−1/s) edges. We construct graphs matching this bound with t≈9s, improving on factorial-type bounds. In this talk, I will explain probabilistic and geometric ideas behind the construction. |
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26.02.40686 Filip Jasiński |
Podstawy Informatyki A Universal Skolem Set of Positive Lower by Density Florian Luca, Joël Ouaknine and James Worrell |
| The Skolem Problem asks to decide whether a given integer linear recurrence sequence (LRS) has a zero term. Decidability of this problem has been open for many decades, with little progress since the 1980s. Recently, a new approach was initiated via the notion of a Skolem set – a set of positive integers relative to which the Skolem Problem is decidable. More precisely, S is a Skolem set for a class L of integer LRS if there is an effective procedure that, given an LRS in L, decides whether the sequence has a zero in S. A recent work exhibited a Skolem set for the class of all LRS that, while infinite, had density zero. In the present work we construct a Skolem set of positive lower density for the class of simple LRS. |
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22.07.24376 Grzegorz Gawryał |
Optymalizacja Kombinatoryczna On topological aspects of orientations |
Constrained graph orientation problem deals with directing graph edges such that graph vertices fulfills some conditions. Here, we are focusing on contant indegree orientations of maximal planar and similar classes of graphs. We analyse the relationship between such orientations and other combinatorial properties of these graphs, including the existence of particular decompositions into trees given by the famous Nash William's theorem. |
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27.03.24364 Rafał Kilar |
Optymalizacja Kombinatoryczna Minimal Non-Two-Colorable Hypergraphs and Minimal Unsatisfiable Formulas |
It is known that the number of edges in a minimal non-2-colorable hypergraph is at least as high as the number of its vertices. We show the link between this and the fact that a minimal unsatisfiable CNF formula with n variables must contain at least n + 1 clauses. We show different proof of these facts and give infinite versions. We also analyze the structure of minimal unsatisfiable CNF formulas with exactly n variables and n + 1 clauses. |
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08.06.21630 László Végh London School of Economics |
Informatyka Teoretyczna Interior point methods are not (much) worse than Simplex |
Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound O(2n n1.5 log n) for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations. The number of iterations of our algorithm is at most O(n1.5 log n) times the number of segments of any piecewise linear curve in the wide neighbourhood of the central path. In particular, it matches the number of iterations of any path-following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the ‘max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. This is joint work with Xavier Allamigeon (INRIA/Ecole Polytechnique), Daniel Dadush (CWI Amsterdam), Georg Loho (U Twente), and Bento Natura (LSE/Georgia Tech). |
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22.10.21520 Katarzyna Król |
Podstawy Informatyki Universal Skolem Sets by Florian Luca, Joel Ouaknine, and James Worrell |
| It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences, namely whether a given such sequence has a zero term. In this paper we introduce the notion of a Universal Skolem Set: an infinite subset S of the positive integers such that there is an effective procedure that inputs a linear recurrence sequence u = (u(n))n≥0 and decides whether u(n) = 0 for some n ∈ S . The main technical contribution of the paper is to exhibit such a set |
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17.03.5211 Szymon Salabura |
Optymalizacja Kombinatoryczna Farey sequence and Graham’s conjectures |
The Farey sequence Fn is the set of rational numbers a/b with 0 ≤ a ≤ b ≤ n and gcd(a,b) = 1. In 1970, Graham proposed the following conjecture. Let a1, a2, ..., an be distinct positive integers. There exist indices i ≠ j, such that we have ai/gcd(ai,aj) ≥ n. In the paper, the authors show interesting properties of Farey sequence sets and how they are closely related to Graham's problems. |
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20.11.5198 Katarzyna Kępińska |
Optymalizacja Kombinatoryczna Color-Critical Graphs on a Fixed Surface |
A graph G is k-color-critical if G is not (k-1)-colorable, but every proper subgraph is. For S, an orientable surface other than the sphere, there are infinitely many k-color-critical graphs if and only if 2<k<6. For k>4 there is the polynomial algorithm for deciding if a graph can be colored with k colors. In this paper, the authors prove those theorems and show some results for list coloring. |
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27.03.5148 Tomasz Mazur, Katzper Michno |
Constrained Backward Time Travel Planning is in P |
| Tematem rozważań będą sieci transportowe modelowane przez dynamiczne grafy, w których wierzchołkach dopuszczalne jest cofanie się w czasie, przy czym nie można cofnąć się o więcej niż pewną liczbę jednostek oraz jest ono obarczone kosztem wyrażonym pewną funkcją kosztu. Skupiamy się na dynamicznych grafach będącymi podgrafami ścieżki. W szczególności podajemy algorytmy wielomianowe dla różnych wariantów szukania trasy z jednego wierzchołka do drugiego minimalizującej w pierwszej kolejności opóźnienie (różnicę między czasem dotarcia a wyruszenia), a drugiej sumaryczny koszt cofania się w czasie. Warianty różnią się ograniczeniami na to, jak możemy cofać się w czasie. Badamy wpływ wyboru funkcji kosztu cofania na problem obliczania optymalnej trasy oraz podajemy warunki konieczne dla funkcji kosztu, aby optymalna trasa istniała. Na koniec podajemy optymalny algorytm on-line na szukanie optymalnej trasy dla funkcji kosztu będącej identycznością, w przypadku, gdy możemy cofać się dowolnie daleko w czasie. |
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20.03.84602 Małgorzata Sulkowska Wrocław University of Technology |
Informatyka Teoretyczna Modularity of minor-free graphs |
Modularity is a well-established parameter measuring the presence of community structure in the graph. It was introduced by Newman and Girvan in 2004. Nowadays it is widely used as a quality function for community detection algorithms. The popular heuristic clustering algorithms (e.g., Louvain algorithm or Leiden algorithm) find a partition using modularity-based approach. We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, for every ε>0, the modularity of any graph in the class with sufficiently many edges is at least 1−ε. This completes the classification of maximally modular classes among all commonly considered subclasses of nowhere dense graphs with maximum degree sublinear in the number of edges.
Joint work with Michał Lasoń |
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01.08.84492 Tomasz Buczyński |
Podstawy Informatyki The Variable Containment Problem by Stefan Kahrs |
The essentially free variables of a term t in some lambda calculus $FV(t)$ form the set $\{x : \forall t =_{beta} u \rightarrow x\in FV(u) \}. This set is signicant once we consider equivalence classes of \lambda terms rather than \lambda terms themselves as for instance in higher order rewriting. An important problem for (generalised) higher order rewrite systems is the variable containment problem. This property is important when we want to consider $t \rightarrow u$ as a rewrite rule and keep n-step rewriting decidable. Variable containment is in general not implied by $FV(t) \supseteq FV(u)$. We give a decision procedure for the variable containment problem of the second order fragment of $\lambda^\rightarrow$. For full $\lambda^\rightarrow$ we show the equivalence of variable containment to an open problem in the theory of PCF; this equivalence also shows that the problem is decidable in the third order case. |
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26.12.68182 Filip Konieczny |
Optymalizacja Kombinatoryczna Factorizing regular graphs |
A q-factor of a k-regular graph is its q-regular subgraph covering all vertices. q-factorization is a partition of edges of a graph into disjoint q-factors. For q-factorization to exist it is necessary that q\mid k. It was proven that for even q the converse is also true - qd-regular graph has a q-factorization. The paper investigates when qd-regular graph with odd q admits q-factorization, given additional assumptions like planarity and/or high connectivity. |
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31.08.68170 Hubert Dej |
Optymalizacja Kombinatoryczna On the Gap Structure of Sequences of Points on a Circle |
The problem of determining a sequence of points on the unit circle is considered, such that at any time t the lengths of the segments (sticks) resulting from splitting the circle at the locations set by the first t points are as equal as possible. The authors consider the sequence xk=lg(2k-1) mod 1 discovered and analyzed by De Brujin and Erdos in 1949 called the log stick-breaking strategy, proven to be optimal under 3 selected measures. The analysis of this sequence is extended by showing an interpretation in which log stick-breaking is a uniquely optimal strategy, and a more general framework is designed in which the optimality of this strategy can be explored. |
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06.01.68120 Dominik Chmura, Jan Klimczak |
Derandomized Squaring of Graphs |
Praca opisuje "zderandomizowany" odpowiednik podnoszenia grafu do kwadratu. Nowa operacja zwiększa spójność grafu (mierzoną jako druga co do wielkości wartość własna macierzy sąsiedztwa) prawie tak dobrze jak potęgowanie grafu, zwiększając stopień grafu nie kwadratowo, a jedynie o stałą. Przedstawiono również kilka zastosowań tej konstrukcji, m.in. algorytm alternatywny do wyniku O. Reingolda, który pozwala deterministycznie badać osiągalność w grafach nieskierowanych w logarytmicznej pamięci. |
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12.11.65436 Sophie Spirkl University of Waterloo |
Informatyka Teoretyczna Induced subgraphs and treewidth: H-free graphs |
Treewidth is an important measure of the “complexity” of a graph, and as part of the Graph Minors project, Robertson and Seymour characterized unavoidable subgraphs of graphs with large treewidth. Here we are interested in unavoidable induced subgraphs instead. In this context, Lozin and Razgon characterized all finite families F of graphs such that F-free graphs have bounded treewidth. I will talk about related result, characterizing which graphs H have the property that excluding H as well as four families of large treewidth (a complete graph, a complete bipartite graph, all subdivisions of a wall, and their line graphs) as induced subgraphs leads to a class of bounded treewidth. Joint work with Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, and Sepehr Hajebi |
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28.03.65327 Piotr Kubaty |
Podstawy Informatyki Decision Problems for Second-Order Holonomic Recurrences by Eike Neumann, Joel Ouaknine and James Worrel |
We study decision problems for sequences which obey a second-order holonomic recurrence of the form $f (n + 2) = P (n)f (n + 1) + Q(n)f (n)$ with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P . We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f (1), f (2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones. |
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21.08.49017 Kamil Galewski |
Optymalizacja Kombinatoryczna Majority colorings of sparse digraphs |
A Majority k-coloring of a directed graph is an assignment of k colors to its vertices in such a way that every vertex has the same color as at most half of its out-neighbors. It is known that every digraph is majority 4-colorable, but it remains an open question whether every digraph is majority 3-colorable. The authors of the paper validate this conjecture for digraphs with a chromatic number at most 6 and digraphs with a dichromatic number at most 3. They also prove analogous theorems for list coloring: digraphs with a list chromatic number at most 6 or list dichromatic number at most 3 are majority 3-choosable. The paper also investigates which digraphs are majority 2-colorable: the authors show that digraphs without directed odd cycles are majority 2-colorable, but in general deciding whether a given digraph is majority 2-colorable is NP-complete. The last result proposed in this paper is proof that every digraph has a fractional majority of 3.9602-coloring. |
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26.04.49005 Piotr Kaliciak |
Optymalizacja Kombinatoryczna A counterexample to the lights out problem |
In the basic Lights Out problem, we are given the undirected graph of turned-off lights, and our goal is to turn on all the lights. In the generalized version of this problem, our mission is to assign every vertex a value from 0 to p, such that for every vertex, the sum of values in its neighbors is equal to 0 mod p. The authors not only prove that a generalized version of this problem isn't always solvable but also they show conditions, under which the problem has a solution. |
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31.08.48954 Bartłomiej Błoniarz, Hubert Dej |
More on Change-Making and Related Problems |
Mając do dyspozycji zbiór n typów monet o wartościach całkowitych oraz wartość docelową t, w problemie wydawania reszty (change-making) szukamy minimalnej liczby monet, które sumują się do t, zakładając możliwość wykorzystania dowolnej liczby monet każdego typu. W bardziej ogólnej wersji tego problemu (w wersji all-targets), chcemy obliczyć wyniki dla wszystkich wartości docelowych 0, 1, ..., t. Klasyczny algorytm dynamiczny rozwiązuje ten problem w czasie O(nt). W publikacji autorzy przedstawiają szereg nowych wyników dotyczących problemu wydawania reszty i innych pokrewnych problemów. Dla u – wartości największej z monet (wagi najcięższego przedmiotu w przypadku problemu plecakowego) pokażemy algorytmy o złożoności: |
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08.07.46271 Hoang La Jagiellonian |
Informatyka Teoretyczna On Barnette's Conjecture for directed graphs |
Knauer and Valicov showed that multiples conjectures from seemingly different problems all fit into the same framework of cuts in matchings of 3-connected cubic graphs. They unite Tait's, Barnette's, and Tutte's conjectures on Hamiltonicity in cubic graphs, Neumann-Lara's on the dichromatic number of planar graphs, and Hochstättler's on contraction of even digraphs. More precisely, these are all equivalent to conjectures of the form ''Every 3-connected, cubic, bipartite/planar/directed graph contains a perfect matching without (directed) cut''. If you drop two of these restrictions (bipartite, planar, directed), then the conjecture is false. If you drop one or none, then the conjecture remains open. We are investigating the dual version of the conjecture with all three restrictions, namely ''Every directed planar Eulerian triangulation can be vertex-partitioned into two acyclic sets''. This new framework can be useful as a planar Eulerian triangulation has an unique partition into three independent sets. |
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20.11.46161 Aleksander Katan |
Podstawy Informatyki The combinator M and the Mockingbird lattice by Samuele Giraudo |
| We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator M. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule Mx_1 → x_1x_1. We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on M and that all connected components of its rewrite graph are Hasse diagram of lattices. This last result is based on the introduction of new lattices on duplicative forests, which are sorts of treelike structures. These lattices are not graded, not self-dual, and not semidistributive. We present some enumerative properties of these lattices like the enumeration of their elements, of the edges of their Hasse diagrams, and of their intervals. These results are derived from formal power series on terms and on duplicative forests endowed with particular operations. |
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15.04.29852 Rafał Pyzik |
Optymalizacja Kombinatoryczna Every graph contains a linearly sized induced subgraph with all degrees odd |
It was proven by Gallai, that every undirected graph on n vertices contains an induced subgraph on at least n/2 vertices with all degrees even. It is natural to ask a similar question for odd degrees. It was conjectured, that in every graph on n vertices, without isolated vertices, we can find an induced subgraph on at least cn vertices with all degrees odd for some constant c>0. We will prove this conjecture for c=1/10000. |
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20.12.29839 Justyna Jaworska |
Optymalizacja Kombinatoryczna The Lovász Local Lemma is Not About Probability |
Since the original statement of Lovas Local Lemma in 1973, multiple variants of the lemma with different levels of complexity have been formulated. We will present a general theorem from which most known variants of LLL follow. Additionally, the results will be generalized to supermodular functions rather than probability measures, allowing a wider range of applications. |
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03.03.27106 Wojciech Czerwiński University of Warsaw |
Informatyka Teoretyczna Reachability problem in Vector Addition Systems |
Recently we managed with co-authors to settle the complexity of the reachability problem for Vector Addition Systems (VASes) to be Ackermann-complete. Despite of that the combinatorics of VASes still remains mysterious and there is a bunch of very natural problems about which we know shockingly little. The focus of my talk will be on tools. I will present techniques, which led to the proof of Ackermann-hardness for the reachability problem and which hopefully may help in solving the remaining challenges. |
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09.12.10686 Jędrzej Kula |
Optymalizacja Kombinatoryczna Complete minors and average degree – a short proof |
We call graph H a minor of graph G, if there exists such a sequence of deletions of vertices, deletions of edges, or contradictions of edges, which transforms G into H. The authors of the paper created a short proof of the result of Kostochka and of Thomasen. The proven theorem states that for every graph whose vertices have the average degree d the graph itself also contains a complete minor of order Ω(d/sqrt(log(d))). |
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14.08.10674 Krzysztof Ziobro |
Optymalizacja Kombinatoryczna Note on the Lamp Lighting Problem |
In the most basic version of the Lamp Lighting Problem, we are given an undirected graph G. We can toggle light in a chosen vertex and all of its neighbors. Our goal is to decide if it is possible to turn on the light in all vertices by performing only moves as described. Authors prove that it is always possible and explore other variants of the problem such as the directed case or the problem of checking if all lighting configurations are possible to achieve. |
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26.10.7940 Jędrzej Hodor Jagiellonian |
Informatyka Teoretyczna Dimension of planar posets |
It is a longstanding open problem if posets with a planar cover graph are dim-bounded (meaning that large dimension yields a large standard example as a subposet). This notion is the posets' counterpart of the well-studied χ-boundedness in the graph theory. In my talk, I will focus on summarizing the new progress in this area. The dim-boundedness was recently proved for posests with planar diagram and for posets with planar cover graph and a zero. I will try to sketch some ideas standing behind these results. The other interesting related question in the area is the following. Suppose that a planar poset has a large standard example as a subposet, then, how does this standard example look like? There are two canonical constructions of planar posets with large standard example contained, namely, Kelly's example and Trotter's wheel. We believe that these are (in a structural sense) the only ways to draw a standard example on the plane. For example, we proved that a poset with a planar cover graph, a zero, and large dimension contains a large Trotter's wheel. The list of coauthors of substantial results that are going to be discussed in my talk: P.Micek, M.Seweryn, H.S.Blake, W.T.Trotter |
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20.04.76384 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna On a Problem of Steinhaus |
Let N be a positive integer. A sequence X=(x1,x2,…,xN) of points in the unit interval [0,1) is piercing if {x1,x2,…,xn}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most 74 elements. This was later improved to the best possible value of 17 by Warmus, and independently by Berlekamp and Graham. We study a more general variant of piercing sequences. Let f(n)≥n be an infinite nondecreasing sequence of positive integers. A sequence X=(x1,x2,…,xf(N)) is f-piercing if {x1,x2,…,xf(n)}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. A special case of f(n)=n+d, with d a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each d≥0, the maximum length of any (n+d)-piercing sequence is finite. Expressing this maximum length as s(d)+d, they obtained an exponential upper bound on the function s(d), which was later improved to s(d)=O(d3) by Graham and Levy. Recently, Konyagin proved that 2d⩽s(d)<200d holds for all sufficiently big d. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function s(d) satisfies ⌊c1d⌋⩽s(d)⩽c2d+o(d), where c1=ln2/(1−ln2)≈2.25 and c2=(1+ln2)/(1−ln2)≈5.52. We also prove that there exists an infinite f-piercing sequence with f(n)=γn+o(n) if and only if γ≥1/ln2≈1.44. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Rafał Pyzik, and Mariusz Zając. |
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26.08.76333 Łukasz Selwa, Juliusz Wajgelt |
Token sliding on graphs of girth five |
Intuicyjnie problem Token Sliding możemy rozumieć jako grę, w której otrzymujemy graf oraz żetony ustawione na jego wierzchołkach. Pytamy, czy da się uzyskać zadany stan końcowy poprzez przesuwanie żetonów wzdłuż krawędzi grafu tak, że w żadnym momencie dwa żetony nie łączyła wspólna krawędź. Formalnie mamy na wejściu graf G oraz zbiory niezależne wierzchołków Is, It i chcemy stwierdzić czy istnieje sekwencja I1, …, Is zbiorów niezależnych w G taka, że I1 = Is, Il = It oraz Ii ∆ Ii+1 = {u, v} \in E(G). Wykazano wcześniej, że dla grafów o talii (ang. girth) 4 lub mniejszej problem Token Sliding jest W[1]-trudny. Prezentujemy dowód z pracy „Token sliding on graphs of girth five”, że dla grafów o talii 5 lub większej problem Token Sliding jest fixed-parameter tractable (FPT). |
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02.07.73650 Dömötör Pálvölgyi Eötvös Loránd University |
Informatyka Teoretyczna At most 3.55^n stable matchings |
We improve the upper bound for the maximum possible number of stable matchings among n jobs and n applicants (formerly known as n men and n women) from 131072n to 3.55n. To establish this bound, we state a novel formulation of a certain entropy bound that is easy to apply and may be of independent interest in counting other combinatorial objects. Joint work with Cory Palmer |
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15.11.73540 Julian Leśniak |
Podstawy Informatyki Tight rank lower bounds for the Sherali–Adams proof system by Stefan Dantchev, Barnaby Martin and Mark Rhodes |
We consider a proof (more accurately, refutation) system based on the Sherali–Adams (SA) operator associated with integer linear programming. If F is a CNF contradiction that admits a Resolution refutation of width k and size s, then we prove that the SA rank of F is ≤ k and the SA size of F is \leq (k + 1)s + 1. We establish that the SA rank of both the Pigeonhole Principle PHP_n^{n-1} and the Least Number Principle LNP_n is n − 2. Since the SA refutation system rank simulates the refutation system of Lovász–Schrijver without semidefinite cuts (LS), we obtain as a corollary linear rank lower bounds for both of these principles in LS. |
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14.12.57218 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna A Note on Generalized Majority Colorings |
A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an online analog of list coloring. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Mariusz Zając. |
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20.04.57168 Julia Biały, Zofia Glapa |
All Paths Lead to Rome |
Roma to łamigłówka rozgrywana na składającej się z kwadratowych pól planszy rozmiaru n x n. Pola pogrupowane są w obszary składające się z co najwyżej 4 sąsiadujących ze sobą komórek, z których każda albo jest wypełniona, albo ma zostać wypełniona strzałką w jednym z 4 kierunków. Celem gry jest wypełnienie wszystkich komórek strzałkami tak by w każdym obszarze była co najwyżej jedna strzałka w danym kierunku i by podążając zgodnie ze strzałkami można było dojść do wyróżnionego pola Roma z każdego pola na planszy. Autorzy pracy rozważają złożoność obliczeniową gry i pokazują, że uzupełnienie planszy zgodnie z zasadami jest problemem NP-zupełnym, zliczenie możliwych rozwiązań jest #P zupełne oraz wyznaczenie liczby zadanych z góry strzałek koniecznych by gra miała tylko jedno rozwiązanie jest ΣP2 - zupełne. Praca dowodzi też, że zakładając prawdziwość ETH problem uzupełnienia planszy dla danej instancji gry nie może być rozwiązany w czasie O(2o(n)). Omawia także algorytm programowania dynamicznego rozwiązujący planszę gry, opierający się na strukturach Catalana. |
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24.02.54485 Vida Dujmović University of Ottawa |
Informatyka Teoretyczna Stack and Queue layouts |
This talk will focus on two graph parameters: stack layouts (aka. book embeddings) and queue layouts of graphs. I will talk about the history of these two graph parameters, their still not fully understood relationship and some recent breakthroughs. |
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11.07.54375 Juliusz Wajgelt |
Podstawy Informatyki Short Proofs of Normalization for the simply-typed λ-calculus, permutative conversions and Godel’s T by Felix Joachimski and Ralph Matthes |
| Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style `a la Tait and Girard. It is also shown that the extension of the system with permutative conversions by (\eta) rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of “immediate simplification”. By introducing an infinitely branching inductive rule the method even extends to Godel’s T |
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08.08.38053 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Recoloring Unit Interval Graphs with Logarithmic Recourse Budget |
We study the problem of coloring a unit interval graph that changes dynamically. In our model the unit intervals are added or removed one at a time and have to be colored immediately so that no two overlapping intervals share the same color. After each update, only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper, we show, that if the graph remains k-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of O(k7logn) while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bosek et al., Recoloring Interval Graphs with Limited Recourse Budget. SWAT 2020] in terms of both k and n. We complement this result by showing the lower bound of Ω(n) on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As a byproduct of independent interest, we include a new result on coloring proper circular-arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs A. We show that if there is a set A′ of non-intersecting unit arcs of size L2−1 such that A∪A′ does not contain L+1 arcs intersecting in one point, then it is possible to color A with L colors. This complements the work on unit circular arc coloring, which specifies sufficient conditions needed to color A with L+1 colors or more. This is joint work with Anna Zych-Pawlewicz. |
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21.10.35319 Friedrich Eisenbrand École Polytechnique Fédérale de Lausanne |
Informatyka Teoretyczna Integer programming with few constraints |
The talk features a survey as well as recent new results on two independent approaches to derive efficient algorithms for integer programming, namely algorithms based on the geometry of numbers and dynamic programming techniques, with an extra spotlight on the case in which the number of constraints (apart from bounds on the variables) is small. We will highlight open problems and possible future directions. The presented results of the speaker have been jointly achieved with Daniel Dadush, Thomas Rithvoss and Robert Weismantel. |
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05.03.35210 Mateusz Olszewski |
Podstawy Informatyki Implicit computation complexity in higher-order programming languages (A Survey in Memory of Martin Hofmann) by Ugo Dal Lago |
| This paper is meant to be a survey about implicit characterizations of complexity classes by fragments of higher-order programming languages, with a special focus on type systems and subsystems of linear logic. Particular emphasis will be put on Martin Hofmann’s contributions to the subject, which very much helped in shaping the field. |
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02.04.18888 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Dimension of planar posets |
It is a long-standing open problem if planar posets are dim-bounded (an analog of chi-bounded in the graph theory). I summarize recent progress on this problem. We explore different notions of what does it mean for posets to be planar. Finally, I will sketch the proof of dim-boundedness in the case of posets with planar cover graphs and a zero. |
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15.06.16154 Paul Seymour Princeton University |
Informatyka Teoretyczna Getting closer to the Erdős-Hajnal conjecture |
A general n-vertex graph may not have a clique or stable set larger than O(log n), but excluding an induced subgraph makes a significant difference. The Erdős-Hajnal conjecture (from 1977) says that for every graph H, there exists c such that every H-free graph G (that is, not containing H as an induced subgraph) has a clique or stable set of size at least |G|c. This is still open, and is notoriously intractable.
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11.03.43533 Piotr Micek Jagiellonian |
Informatyka Teoretyczna Boolean dimension and dim-boundedness of posets with a unique minimal element whose cover graphs are planar |
In 1989, Nešetřil and Pudlák posed the following challenging question: Do planar posets have bounded Boolean dimension? We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most 13. As a consequence, we are able to show that there is a reachability labeling scheme with labels consisting of O(log n) bits for planar digraphs with a single source. The best known scheme for general planar digraphs uses labels with O(log2n) bits [Thorup, JACM 2004], and it remains open to determine whether a scheme using labels with O(log n) bits exists. The Boolean dimension result is proved in tandem with a second result showing that the dimension of a poset with a planar cover graph and a unique minimal element is bounded by a linear function of its standard example number. However, one of the major challenges in dimension theory is to determine whether dimension is bounded in terms of standard example number for all posets with planar cover graphs. |
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01.10.27105 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna The 1/3 - 2/3 conjecture |
A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field. |
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04.11.24367 Michał Wrona Jagiellonian |
Informatyka Teoretyczna Local consistency methods in Solving CSPs and CSP-like problems over omega-categorical structures |
Feder-Vardi conjecture has been settled independently by Dmitriy Zhuk and Andrei Bulatov. What is perhaps even more interesting, though, is that they not only confirmed the complexity (Feder-Vardi) conjecture, i.e., CSP(B) for a finite structure B is either in P or it is NP-complete, but they also confirmed the algebraic dichotomy conjecture describing tractable B in terms of operations preserving B. A similar algebraic dichotomy conjecture called an infinite algebraic dichotomy conjecture has been established for CSP(B) over first-order reducts B of finitely bounded homogeneous structures, all of which are in particular omega-categorical. Despite recent advances towards solving this dichotomy, it still seems to be wide open. One of the reasons is probably that local consistency and similar algorithmic techniques are in this context not yet fully understood. This step seems to be crucial since the characterization of finite-domain CSP solvable by local consistency is considered as a major step towards the resolution of the dichotomy. In this talk, I will survey the results on the local consistency methods in solving CSP and CSP-like problems over omega-categorical structures. |
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29.04.24258 Karolina Gontarek |
Podstawy Informatyki THE TU–DENG CONJECTURE HOLDS ALMOST SURELY by LUKAS SPIEGELHOFER AND MICHAEL WALLNER |
The Tu–Deng Conjecture is concerned with the sum of digits w(n) of n in base 2 (the Hamming weight of the binary expansion of n) and states the following: assume that k is a positive integer and t \in {1, . . . , 2^k − 2}. Then #\{ (a, b) ∈ {0, . . . , 2k − 2}^2 : a + b ≡ t mod 2^k − 1, w(a) + w(b) < k \} \leq ≤ 2^{k-1}
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03.09.7963 Krzysztof Pióro |
Optymalizacja Kombinatoryczna Brooks' Theorem via the Alon-Tarsi Theorem |
Brooks' Theorem states that every connected graph G with maximum degree d is d-colorable unless G is an odd cycle or a complete graph. It is one of the most famous theorem on graph colorings. In the paper, the author presents yet another proof of this theorem. This proof is based on Alon-Tarsi Theorem and it remains valid in a more general choosability version of Brooks' theorem. |
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26.05.7940 Demian Banakh |
Optymalizacja Kombinatoryczna Separating polynomial χ-boundedness from χ-boundedness |
A class of graphs is hereditary χ-bounded if it is closed under taking induced subgraphs and every graph’s chromatic number is bounded by some function of its clique number. A well-known recently stated open question has been whether for every hereditary χ-bounded class that function can be chosen to be a polynomial. We provide a counterexample for it; namely, for any function f, we construct a hereditary χ-bounded class containing graphs of large chromatic number. In particular, for any polynomial f, such a class exists, which answers the aforementioned question negatively. |
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22.08.7885 Jędrzej Kula, Maciej Nemś |
Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs |
Grafy przecięć geometrycznych to grafy, gdzie wierzchołki odpowiadają figurom geometrycznym w d-wymiarowej przestrzeni euklidesowej. Mogą do to być na przykład kule, kwadraty, hiperkostki. Krawędź między dwoma wierzchołkami istnieje, jeśli dwie figury przecinają się. Jest to typowy sposób modelowania na przykład komunikacji bezprzewodowej. W pracy autorzy zajmują się obliczaniem średnicy tego typu grafów. Dokładniej rozważają to, czy da się ten problem rozwiązać w czasie poniżej kwadratowym względem liczby wierzchołków. Na referacie zostanie pokazany dowód algorytmu o czasie działania O(n logn) dla sprawdzania, czy średnica jest mniejsza bądź równa 2 dla grafów przecięć kwadratów jednostkowych równoległych do osi. Następnie zostanie pokazane dolne ograniczenie szukania średnicy dla kul jednostkowych na bazie Orthogonal Vectors Hypothesis. Ograniczenie to pokazuje, że nie ma algorytmów pod kwadratowych przy założeniu Orthogonal Vectors Hypothesis. |
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22.12.5092 Juliusz Wajgelt |
Podstawy Informatyki EFFICIENT FULL HIGHER-ORDER UNIFICATION by PETAR VUKMIROVIC, ALEXANDER BENTKAMP, AND VISA NUMMELIN |
| We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a nite complete set of uni ers. We identify a new such fragment and describe a procedure for computing its uni ers. Our uni cation procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure. |
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09.05.73673 Bartosz Podkanowicz |
Optymalizacja Kombinatoryczna Digraphs are 2-weight choosable |
Consider following problem. We are given a digraph. For every edge, there are 2 options to choose a weight for this edge. We want to pick the weights of edges in a specific way. After picking weights we color vertices. The color of the vertex will be the sum of incoming edges minus the sum of outgoing edges from that vertex. We show that it is always possible to choose weights of edges such that the resulting coloring will be proper. This property is called 2-weight-choosability. |
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30.01.73650 Łukasz Selwa |
Optymalizacja Kombinatoryczna A better lower bound on average degree of 4-list-critical graphs |
A graph G is k-list-critical if it is not (k-1)-choosable, but every proper subgraph of G is (k-1)-choosable. We give a new lower bound for the average degree of incomplete k-list-critical graphs and online k-list-critical graphs. The presented bound improves the earlier known lower bounds for k = 4,5,6.
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29.04.73595 Grzegorz Gawryał, Rafał Kilar |
Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves |
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05.03.70912 Szymon Toruńczyk University of Warsaw |
Informatyka Teoretyczna Ordered graphs of bounded twin-width and monadically NIP graph classes |
We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences.First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Those results are joint work with Bonnet, Giocanti, Ossona de Mendez, Simon, Thomasse, accepted to STOC'22. Time permitting, I will also discuss the more general landscape of monadically NIP graph classes, generalizing both nowhere dense classes and classes of bounded twin-width. |
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29.08.70802 Vitaliy Mysak |
Podstawy Informatyki An Introduction to the Clocked Lambda Calculus byJörg Endrullis, Dimitri Hendriks, Jan Willem Klop, and Andrew Polonsky |
| We give a brief introduction to the clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps. In contrast to the classical λ-calculus, this extension is infinitary strongly normalising and infinitary confluent. The infinitary normal forms are enriched Lévy–Longo Trees, which we call clocked Lévy–Longo Trees. |
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03.01.54508 Rafał Kilar |
Optymalizacja Kombinatoryczna Lower Bounds on the On-line Chain Partitioning of Semi-orders with Representation |
An online chain partitioning algorithm is presented with one element of a poset at a time and has to assign it to a chain, partitioning the poset. We consider posets with elements represented by unit length intervals. The paper slightly improves the lower bound for the minimum number of chains needed by an online algorithm to partition these posets from ⌊3/2 w⌋ to ⌈3/2 w⌉. |
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24.09.54484 Krzysztof Potępa |
Optymalizacja Kombinatoryczna Unit-Monge matrices and seaweed braids |
Simple unit-Monge matrices form a special subclass of square matrices, which can be represented implicitly in linear space by permutations. Somewhat surprisingly, the subclass is closed under distance multiplication. We will show connection between simple unit-Monge matrices and seaweed braids: braids in which each pair of strings crosses at most once. In particular, distance multiplication is equivalent to a "combing procedure", where double-crossings in braid are removed. We will discuss applications of these methods to a few subsequence problems. In particular, the combing procedure can be exploited to obtain an elegant algorithm for all-substring LCS problem. |
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22.12.54429 Andrii Orap, Maksym Zub |
Grundy Distinguishes Treewidth from Pathwidth |
Strukturalne parametry grafów, takie jak treewidth, pathwidth i clique-width, są głównym tematem badań sparametryzowanej złożoności. Głównym celem badań w tej dziedzinie jest zrozumienie „ceny ogólności” tych szerokości: kiedy przechodzimy od pojęć bardziej restrykcyjnych do bardziej ogólnych, jakie są problemy, w których złożoność pogarsza się z fixed-parameter tractable do intractable? Ten rodzaj pytania jest już bardzo dobrze zbadany, ale, co jest dość uderzające, algorytmiczna granica między dwoma (prawdopodobnie) najbardziej centralnymi pojęciami szerokości (notacjami), treewidth i pathwidth, jest nadal niezrozumiała: obecnie nie jest znany żaden naturalny problem na grafie, który byłby W-trudny dla jednego, ale FPT dla drugiego. Rzeczywiście, zaskakującym rozwojem ostatnich kilku lat była obserwacja, że: w przypadku wielu najbardziej paradygmatycznych problemów ich złożoność dla tych dwóch parametrów w rzeczywistości dokładnie się pokrywają, pomimo faktu, że szerokość drzewa jest parametrem o wiele bardziej ogólnym. W ten sposób wydaje się, że dodatkowa ogólność szerokości drzewa nad szerokością ścieżki często przychodzi „za darmo”. Naszym głównym wkładem w ten artykuł jest odkrycie pierwszego naturalnego przykładu, w którym ta ogólność ma wysoką cenę. Rozważamy Grundy Coloring, wariację kolorowania, w której próbujemy obliczyć najgorsze możliwe kolorowanie, które można przypisać do grafu przez zachłanny algorytm First-Fit . Pokazujemy, że ten dobrze zbadany problem jest parametryzowany (FPT) przez pathwidth; jednakże to staje się znacznie trudniejsze (W[1]-hard), gdy jest sparametryzowany przez treewidth. Ponadto pokazujemy, że Grundy Coloring sprawia, że jest drugi skok złożoności dla bardziej ogólnych szerokości, gdy staje się para-NP-hard dla clique-width. Dlatego Grundy Coloring ładnie oddaje złożoność kompromisów między trzema najlepiej zbadanymi parametrami. Uzupełniając obraz, pokazujemy, że Grundy Coloring jest parametryzowane przez FPT według modular-width. |
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29.10.51746 Rose McCarty University of Warsaw |
Informatyka Teoretyczna Circuit decompositions of group-labelled graphs |
This talk focuses on Eulerian graphs whose arcs are directed and labelled in a group. Each circuit yields a word over the group, and a circuit is non-zero if this word does not evaluate to 0. We give a precise min-max theorem for the following problem. Given a vertex v, what is the maximum number of non-zero circuits in a circuit-decomposition where each circuit begins and ends at v? This is joint work with Jim Geelen and Paul Wollan. Our main motivation is a surprising connection with vertex-minors which is due to Bouchet and Kotzig. |
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23.04.51637 Jan Koscisz |
Podstawy Informatyki Functions-as-constructors higher-order unification: extended pattern unification by Tomer Libal and Dale Miller |
| Unification is a central operation in constructing a range of computational logic systems based on first-order and higher-order logics. First-order unification has several properties that guide its incorporation in such systems. In particular, first-order unification is decidable, unary, and can be performed on untyped term structures. None of these three properties hold for full higher-order unification: unification is undecidable, unifiers can be incomparable, and term-level typing can dominate the search for unifiers. The so-called pattern subset of higher-order unification was designed to be a small extension to first-order unification that respects the laws governing λ-binding (i.e., the equalities for α, β, and η-conversion) but which also satisfied those three properties. While the pattern fragment of higher-order unification has been used in numerous implemented systems and in various theoretical settings, it is too weak for many applications. This paper defines an extension of pattern unification that should make it more generally applicable, especially in proof assistants that allow for higher-order functions. This extension’s main idea is that the arguments to a higher-order, free variable can be more than just distinct bound variables. In particular, such arguments can be terms constructed from (sufficient numbers of) such bound variables using term constructors and where no argument is a subterm of any other argument. We show that this extension to pattern unification satisfies the three properties mentioned above. |
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28.08.35342 Jacek Salata |
Optymalizacja Kombinatoryczna A Short Proof of Nash-Williams' Theorem for the Arboricity of a Graph |
Nash-Williams theorem (tree-packing theorem) is a classical result due to Nash-Williams (1961) that characterizes graphs with k edge-disjoint spanning trees. In the seminar, I will present a short and elegant proof of the theorem. |
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21.05.35319 Kamil Galewski |
Optymalizacja Kombinatoryczna Bears with Hats and Independence Polynomials |
The hat guessing game is a game in which bears sit in the vertices of an undirected graph. A demon puts hats on the bears' heads. Each hat has one of the h available colors. Each bear sees only the hat colors of his neighbors. The goal of the bears is to guess the color of their hats - each bear has g tries to guess his hat color. The bears win if at least one of them has guessed the color of his hat correctly. This paper describes the relationship between the hat guessing game and the independence polynomial of graphs. |
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23.06.32581 Michał Seweryn Jagiellonian |
Informatyka Teoretyczna Forcing walls with divisibility constraints |
An n-wall is a graph obtained from the square grid with n rows and 2n columns by deleting every odd vertical edge in every odd row and even vertical edge in every even row, then deleting the two resulting vertices of degree 1, and finally subdividing each edge any number of times. Thomassen showed that there exists a function f(n,m) such that every f(n,m)-wall contains an n-wall such that every path between two branch vertices has length divisible by m. We study the asymptotics of the optimal such function f(n,m). For odd m we show that f(n,m) = O(n·poly(m)). In the case m=2, we obtain a bound f(n, 2) = O(n·log n). This is joint work with Piotr Micek, Raphael Steiner and Sebastian Wiederrecht. |
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28.10.32471 Roch Wójtowicz |
Podstawy Informatyki SHARP THRESHOLDS OF GRAPH PROPERTIES, AND THE k-SAT PROBLEM by EHUD FRIEDGUT AND AN APPENDIX BY JEAN BOURGAIN |
Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that
A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob ́as and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random k-CNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability. |
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22.04.16177 Szymon Salabura |
Optymalizacja Kombinatoryczna Contact graphs of ball packings |
A contact graph of a ball packing is a graph with non-intersecting balls as vertices and edges between pairs of tangent balls. In the seminar, we will focus on the upper bounds for the average degree of such graphs in any number of dimensions. |
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13.01.16154 Mateusz Pach |
Optymalizacja Kombinatoryczna Exponentially many 3-colorings of planar triangle-free graphs with no short separating cycles |
It has been conjectured that every planar triangle-free graph G has exponentially many proper vertex-3-colorings. In this paper, the conjecture is disproved. It is also shown that the conjecture holds if we add an assumption about the non-existence of separating cycles of lengths 4 and 5. Specifically, it is proved that the number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. |
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11.04.16099 Krzysztof Pióro, Krzysztof Potępa |
Breaking the Cubic Barrier for (Unweighted) Tree Edit Distance |
W problemie odległości między drzewami dane są dwa ukorzenione drzewa z etykietami na krawędziach. Dodatkowo dla każdego wierzchołka jego dzieci mają ustalony porządek. Naszym celem jest znalezienie minimalnej liczby operacji kontrakcji krawędzi i zmiany etykiety krawędzi, tak aby doprowadzić oba drzewa do takiego samego drzewa. Autor pracy pokazuje algorytm o złożoności O(n2.9546) dla wariantu tego problemu, w którym operacje mają koszty jednostkowe. Jest to pierwszy podsześcienny algorytm dla problemu odległości edycyjnej między drzewami. Warto tutaj zwrócić uwagę, że dla wariantu o dowolnych kosztach operacji istnieje warunkowe ograniczenie dolne, które mówi, że nie istnieje dla tego problemu algorytm podsześcienny. Zatem autor pokazuje, że wariant z jednostkowymi kosztami najprawdopodobniej jest istotnie prostszy od wariantu ogólnego. Aby złamać granicę O(n3) autor redukuje problem do mnożenia macierzy max-plus, w których sąsiednie elementy różnią się co najwyżej o stałą. O takim problemie zostało już udowodnione wcześniej, że może zostać rozwiązany w czasie podsześciennym. |
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17.02.13416 Jakub Kozik Jagiellonian |
Informatyka Teoretyczna Deterministic Constructions of 3-Chromatic Hypergraphs with Few Edges |
How many edges do we need to build a k-uniform hypergraph that cannot be properly two colored? Using the probabilistic argument Erdös proved in 1964, that there exist such hypergraphs with roughly k2·2k edges. However, without a random source at hand, the sizes that we can achieve by efficient procedures are much larger. The first and only known explicit construction with 2k+o(k) edges was proposed by Gebauer in 2013. We will discuss how it can be improved first by randomizing and then derandomizing it once more. |
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01.02.79149 Karolina Gontarek |
Optymalizacja Kombinatoryczna On topological aspects of orientations |
The paper considers two classes of planar graphs: maximal planar graphs and maximal bipartite planar graphs. The authors describe how these graphs can be oriented in the way that each vertex has prescribed indegree. Then the relation of such orientations to specific graph decompositions and orderings on the vertex set is provided. Discussed orientations can be used to characterize some of the planar graphs. Described properties have applications e.g. in graph drawing and planar augmentation problems.
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25.10.79125 Ruslan Yevdokymov |
Optymalizacja Kombinatoryczna Flexible Color Lists in Alon and Tarsi’s Theorem, and Time Scheduling with Unreliable Participants |
By describing a winning strategy for Mrs. Correct in the coloring game of Mr. Paint and Mrs. Correct author presents a purely combinatorial proof of a strengthening of Alon and Tarsi's Theorem. Strengthening of the theorem also leads to the strengthening of its applications, for example, upper bounds for list chromatic numbers of bipartite graphs, list chromatic indices of complete graphs, and chess tournament time scheduling problem with unreliable participants. |
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21.01.79071 Jakub Fedak, Mateusz Golonka |
Wordle is NP-hard |
| Wordle jest grą dla jednego gracza, której celem jest zgadnięcie pewnego słowa x wybranego ze słownika D. Aby zgadnąć słowo x gracz może wykonać co najwyżej k prób, przy czym w każdej próbie gracz musi podać słowo, które również znajduje się w słowniku D. Wszystkie słowa w słowniku mają długość n. Po każdej próbie zgadnięcia gracz otrzymuje informację o pozycjach, na których jego słowo zgadza się z rozwiązaniem oraz o literach z podanego słowa, które znajdują się w rozwiązaniu, lecz na innej pozycji. Autorzy udowadniają, że następujący problem jest NP-trudny: mając dany słownik D oraz liczbę naturalną k powiedzieć, czy gracz może zagwarantować zgadnięcie słowa w k próbach. Ponadto autorzy dowodzą, że dla słów długości 5 ten problem pozostaje trudny, a nawet w tym przypadku przybliżenie najmniejszej liczby prób potrzebnej do zagwarantowania zgadnięcia słowa jest NP-trudne. W pracy znajdują się również wyniki dotyczące złożoności parametryzowanej oraz kilka pytań otwartych związanych z tym tematem. |
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28.11.76387 Andrew Suk University of California at San Diego |
Informatyka Teoretyczna Unavoidable patterns in simple topological graphs |
A simple topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by non-self-intersecting arcs connecting the corresponding points, with the property that any two edges have at most one point in common. In 2003, Pach-Solymosi-Tóth showed that every n-vertex complete simple topological graph contains a topological subgraph on m = Ω(log n) vertices that is weakly isomorphic to the complete convex geometric graph or to the complete twisted graph on m vertices. Here, we improve this bound to (log n)1/4−o(1). I will also discuss other related problems as well. This is joint work with Ji Zeng. |
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23.05.76278 Roch Wójtowicz |
Podstawy Informatyki SEMINARIUM i WYSTĄPIENIE ROCHA WÓJTOWICZA PRZENIESINE NA 11.05.2022 |
Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that
A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollob ́as and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random k-CNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability. |
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27.09.59983 Wojciech Buczek |
Optymalizacja Kombinatoryczna On an early paper of Maryam Mirzakhani |
In this seminar, we will talk about Maryam Mirzakhani, who had an enormous influence on research about Combinatorics. We will study her idea of creating a small (with (only!) 63 vertices), non-4-choosable planar graph, which is also 3-choosable. We will also consider other problems she worked on. |
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19.06.59960 Maciej Nemś |
Optymalizacja Kombinatoryczna Avoiding squares over words with lists of size three amongst four symbols |
Word creation from lists of size t is a problem where for alphabet Σ each sign of created word is chosen from a list of t different signs from Σ. Word is "square-free" when it does not contain any squares. A square is a word of form ww with w being a nonempty word. The author first shows that there are at least 2.45n square-free words of length n created from lists of 4. This is an improvement from the previous bound which is 2n. After that, the main result of the paper is shown which is an existence of at least 1.25n words of length n from lists of 3. |
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16.09.59905 Piotr Kaliciak, Kamil Galewski |
A Simple Algorithm for Graph Reconstruction |
| Praca skupia się na efektywnej rekonstrukcji grafu, przy pomocy zapytań o odległości między wierzchołkami. Rozważane grafy są spójne, nieważone oraz mają ograniczony stopień, a celem jest znalezienie wszystkich krawędzi w grafie. Analizowany jest prosty algorytm rekonstrukcji. Autorzy dowodzą, że na ∆-regularnym grafie wykonuje on O(n*polylog(n)) zapytań. Ponadto można go zmodyfikować pod inne rodzaje zapytań. Co więcej, algorytm ten łatwo jest zrównoleglić. W przypadku grafów o ograniczonym stopniu, algorytm potrzebuje o(n2) zapytań. |
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23.07.57222 Sergey Norin McGill University |
Informatyka Teoretyczna Brambles, stack number and topological overlap |
A (strict) bramble in a graph G is a collection of subgraphs of G such that the union of any number of them is connected. The order of a bramble is the smallest size of a set of vertices that intersects each of the subgraphs in it. Brambles have long been part of the graph minor theory toolkit, in particular, because a bramble of high order is an obstruction to existence of a low width tree decomposition. |
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12.02.40795 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna From 1-2-3 conjecture to Riemann hypothesis |
We consider some coloring issues related to the famous Erdős Discrepancy Problem. A set of the form As,k={s,2s,…,ks}, with s,k ∈ N, is called a homogeneous arithmetic progression. We prove that for every fixed k there exists a 2-coloring of N such that every set As,k is perfectly balanced (the numbers of red and blue elements in the set As,k differ by at most one). This prompts reflection on various restricted versions of Erdős' problem, obtained by imposing diverse confinements on parameters s,k. In a slightly different direction, we discuss a majority variant of the problem, in which each set As,k should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in {+1,−1}. In particular, whether there is such a function with partial sums bounded from above. |
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17.03.38057 Alex Scott University of Oxford |
Informatyka Teoretyczna Induced subgraphs of induced subgraphs of large chromatic number |
We prove that for every graph F with at least one edge there is a constant cF and there are graphs H of arbitrarily large chromatic number and the same clique number as F such that every F-free induced subgraph of H has chromatic number at most cF. (Here a graph is F-free if it does not contain an induced copy of F.) This generalizes theorems of Briański, Davies and Walczak, and of Carbonero, Hompe, Moore and Spirkl. We further show an analogous statement where clique number is replaced by odd girth. Joint work with Antonio Girao, Freddie Illingworth, Emil Powierski, Michael Savery, Youri Tamitegama and Jane Tan. |
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14.01.21653 Marcin Serwin |
Optymalizacja Kombinatoryczna Can a party represent its constituency? |
Upon gaining p% votes in an election in a proportional system, a party appoints p% of its proposed candidates to represent the party. The order of candidates to appoint is chosen beforehand. This may create tensions if the party members are not perfectly aligned politically, if some candidates of particular tendency are lower down the list and thus less likely to be appointed. This presentation examines the problem of existence and characterization of the list that would not create such tension and related problems. |
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07.10.21629 Piotr Kaliciak |
Optymalizacja Kombinatoryczna 2-List-coloring planar graphs without monochromatic triangles |
The author is considering a planar graph, where every vertex has a list of 2 colors, and every coloring of this graph has to choose for every vertex one of these two colors. Unlike the standard colorings, the author doesn't mind having a monochromatic edge, but he tries to avoid having a monochromatic triangle. In this paper, he not only proves, that every planar graph can be colored this way, for every list assignment, but also he proves a stronger result for graphs with some vertices pre-colored. |
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04.01.21575 Bartłomiej Błoniarz, Inka Sokołowska |
On Problems Related to Unbounded SubsetSum: A Unified Combinatorial Approach |
| Unbounded SubsetSum to problem w którym dane są liczby c,u oraz n liczb całkowitych w1,...,wn z przedziału [1,u]. Trzeba odpowiedzieć na pytanie czy istnieją liczby całkowite m1,...,mn spełniające c = w1*m1 + ... + wn*mn. W wersji all-target problemu dana jest liczba naturalna t i należy podać odpowiedź dla wszystkich instancji z c z przedziału [0,t]. Praca skupia się na trzech generalizacjach tego problemu: 1. All-target Unbounded Knapsack - wariant dobrze znanego problemu plecakowego, dla którego przedstawiony jest algorytm Õ(T(u)+t) gdzie T(n) to czas obliczania (min,+)-splotu dla tablic długości n 2. All-target CoinChange - wariant problemu wydawania reszty, dla którego przedstawiony jest algorytm Õ(u+t) 3. Residue Table, dla którego przedstawiony jest algorytm Õ(u). |
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10.11.18891 Marcin Briański Jagiellonian |
Informatyka Teoretyczna Separating polynomial χ-boundedness from χ-boundedness and thereabouts |
If a graph contains no large complete subgraph but nonetheless has high chromatic number what can we say about the structure of such a graph? This question naturally leads to investigation of χ-bounded classes of graphs — graph classes where a graph needs to contain a large complete subgraph in order to have high chromatic number. This an active subfield of graph theory with many long standing open problems as well as interesting recent developments. In this talk I will present a construction of a hereditary class of graphs which is χ-bounded but not polynomially χ-bounded. This construction provides a negative answer to a conjecture of Esperet that every χ-bounded hereditary class of graphs is polynomially χ-bounded. The construction is inspired by a recent paper of Carbonero, Hompe, Moore, and Spirkl which provided a counterexample to another conjecture of Esperet. This is joint work with James Davies and Bartosz Walczak (available at arXiv:2201.08814) |
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06.05.18782 Aleksander Katan |
Podstawy Informatyki A simple proof of the undecidability of strong normalization by Paweł Urzyczyn |
| The purpose of this note is to give a methodologically simple proof of the undecidability of strong normalization in the pure lambda calculus. For this we show how to represent an arbitrary partial recursive function by a term whose application to any Church numeral is either strongly normalizable or has no normal form. Intersection types are used for the strong normalization argument. |
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21.09.87362 Katarzyna Król |
Optymalizacja Kombinatoryczna On List-Coloring Outerplanar Graphs |
An outerplanar graf is a planar graph whose vertices can all be drawn on the outer face. The author researched the problem of coloring outerplanar graphs from lists. First, it is shown that the outerplanar graph is L-colorable if satisfies |L(v)| ≥ min{deg(v),4} and is bipartite. Then additional assumptions are searched for so that a similar inequality could define L-colorability in general outerplanar graphs. The results given by the author are the best possible for each condition in the bounds and hypotheses. |
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14.06.87339 Jędrzej Kula |
Optymalizacja Kombinatoryczna Multiple list colouring of planar graphs |
Since every planar graph G can be colored by 4 colors, there is also an integer m such that G is (4m,m)-choosable. The problem here is that such m is changing with G. The author of this paper proves that there cannot be such a universal m that every planar graph is (4m,m)-choosable. To be precise he shows that for each positive integer m, there is a planar graph G which is not (4m+⌊(2m-1)/9⌋,m)-choosable. Also, he poses some conjectures about planar graphs multiple list coloring. |
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09.09.87284 Tomasz Buczyński, Łukasz Gniecki |
On Determinism Versus Non-Determinism and Related Problems |
| Pokazujemy, że dla wielotaśmowych maszyn Turinga działających w czasie liniowym, niedeterminizm jest mocniejszy od determinizmu, czyli że klasa języków rozpoznawanych przez takie maszyny deterministyczne jest ścisłą podklasą języków rozpoznawanych przez takie maszyny niedeterministyczne. |
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18.07.84601 Raphael Steiner ETH Zürich |
Informatyka Teoretyczna New bounds for relatives of Hadwiger's conjecture |
In this talk, I will present some recent results on two variants of Hadwiger's conjecture. First, I will discuss the so-called Odd Hadwiger's conjecture, a strengthening of Hadwiger's conjecture proposed by Gerards and Seymour in 1995. First I will show how, using a simple new trick, one can reduce the problem of coloring graphs with no odd Kt-minor to coloring graphs with no Kt-minor up to a constant factor of 2, thereby improving previous upper bounds for this problem. Then, I will outline how a similar idea can be used to significantly improve the known bounds for clustered colorings of odd Kt-minor free graphs, in which we look for (possibly improper) colorings with monochromatic components of small size. Second, I will discuss the so-called List Hadwiger's conjecture, which states that there exists a constant c such that every graph with no Kt-minor is ct-choosable (i.e., list colorable). I will show a probabilistic construction of a new lower bound 2t-o(t) for list coloring Kt-minor free graphs, this refutes a conjecture by Kawarabayashi and Mohar which stated that one can take c=3/2. I will then show how some well-chosen modifications to our construction can be used to prove lower bounds also for list coloring H-minor free graphs where H is non-complete. In particular, I will show that Ks,t-minor free graphs for large comparable s and t can have list chromatic number at least (1-o(1))(s+t+min(s,t)), this refutes a 2001 conjecture by Woodall. |
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10.01.84492 Filip Synowiec |
Podstawy Informatyki Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability by Antoine Genitrini and Cécile Mailler |
| This article is motivated by the following satisfiability question: pick uniformly at random an and{or Boolean expression of length n, built on a set of $k_n$ Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence. The fundamental breakthrough of this paper is to generalize the previous results for any (reasonable) sequence of integers which enables us, in particular, to solve the above satisfiability question. We also analyze the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new quotient model happens to exhibit a very interesting threshold (or saturation) phenomenon. |
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16.05.68197 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Clustered Coloring and Hadwiger's conjecture |
Hadwiger conjecture states, that for every Ks+1 minor free graph it can be colored with s colors. For now, it is wide open. There are plenty of well-known results improving the bound on the number of colors. However, there exists another approach to make the problem easier. We can relax the notion of proper coloring. A graph class can be η-clustered colored with n colors if in every graph only n colors are used and the size of each monochromatic component is bounded by η. Liu and Wood proved that for a class of graphs excluding Ks+1 minor can be η(s)-clustered colored with s+2 colors, which is almost optimal (s < s+2). I will describe their approach and prove the result in a simplified case. |
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06.02.68174 Grzegorz Gawryał |
Optymalizacja Kombinatoryczna The Catalan matroid |
A path of length 2n, that starts in (0,0) and at each step moves from (x,y) to (x+1,y+1) or (x+1,y-1) is a Dyck path, if it ends in (2n,0) and never passes below y=0 line. Such paths are counted by Catalan numbers. In this presentation, we'll show, that the Dyck paths for fixed n form a matroid. We'll show what are bases, independent sets, and other matroid-related terms in this object, explore some properties of this matroid, and see how it generalizes to shifted matroids. |
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06.05.68119 Mateusz Pach, Michał Wronka |
Making Life More Confusing for Firefighters |
| Problem Firefighter polega na opracowaniu strategii rozsyłania strażaków do obrony wierzchołków grafu przed rozprzestrzeniającym się przez krawędzie ogniem, tak by jak najmniej wierzchołków spłonęło; problem ten jest NP-trudny dla znakomitej większości klas grafów. By zamodelować scenariusz z cywilami pomagającymi strażakom, wprowadzamy problem Temporal Firefighter będący rozszerzeniem na dynamiczne grafy. Pokazujemy, że problem Temporal Firefighter jest NP-trudny i pozostaje taki dla wszystkich klas grafów (poza jedną) o których wiadomo, że posiadają wielomianowe rozwiązanie problemu Firefighter. Pokazujemy też algorytm FPT dla Temporal Firefighter, parametryzowany wartością vertex-interval-membership-width. |
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12.03.65436 Mathieu Mari University of Warsaw and IDEAS-NCBR |
Informatyka Teoretyczna A (2+ε)-Approximation Algorithm for Maximum Independent Set of Rectangles |
We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences. In this talk, I will present a (2+ϵ)-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 4. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to 2+ϵ. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to O(1/ϵ) line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR. At the end of the talk, I will present a bunch of open questions related to the problem.
This is a joint work with Waldo Gálvez, Arindam Khan, Tobias Mömke, Madhusudhan Reddy and Andreas Wiese |
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05.09.65326 Michał Woźny |
Podstawy Informatyki COUNTING WITH IRRATIONAL TILES by SCOTT GARRABRANT and IGOR PAK |
| We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of certain binomial multisums. We then give asymptotic applications and establish connections to hypergeometric functions and Catalan numbers. |
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02.10.49008 Krzysztof Ziobro |
Optymalizacja Kombinatoryczna A note on polynomials and f-factors of graphs |
The factor of a graph is its spanning subgraph which adheres to given constraints on the degrees. The authors of the article discuss the f-factor, which for every vertex defines a set of possible degrees. The main result shows a new sufficient condition for the existence of an f-factor in a given graph. Authors obtain it by using Combinatorial Nullstellensatz. |
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29.12.48953 Ignacy Buczek, Michał Woźny |
Sorting Balls and Water: Equivalence and Computational Complexity |
Problemy sortowania od długiego czasu są obiektem różnego rodzaju badań. Ostatnio dwie gry na telefon w tematyce sortowania zyskały na popularności. W tych grach, gracz ma do dyspozycji urny wypełnione kolorowymi obiektami (w przypadku jednej są to kule, a w przypadku drugiej woda) oraz kilka pustych urn, a jego celem jest posortowanie obiektów zgodnie z kolorami. W jednym ruchu może on przenieść obiekty z jednej urny do drugiej, jeżeli kolor przenoszonych obiektów zgadza się z kolorem najwyższego obiektu docelowej urny lub urna ta jest pusta. W pracy autorzy badają złożoność obliczeniową tych łamigłówek. Na początku pokazują, że te gry są w równoważne pod kątem rozwiązywalności. Dokładniej mówiąc, rozwiązywalność stanu początkowego gry nie zależy od tego czy obiekty zachowują się jak kule, czy jak woda. Dowodzą również, że dla każdej tak-instancji istnieje rozwiązanie wielomianowego rozmiaru, co pokazuje, że problem rozwiązywalności tych łamigłówek jest w NP. Następnie uzasadniają, że ten problem jest NP-zupełny. Znajdują również wielomianowe algorytmy dla szczególnych przypadków. Na samym końcu zastanawiają się, jak wiele pustych urn jest potrzebnych, aby instancja była rozwiązywalna niezależnie od początkowego rozmieszczenia obiektów. Pokazują nietrywialne ograniczenia (dolne i górne) zależne od ilości początkowo zapełnionych urn i ich pojemności. |
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05.11.46270 Marek Sokołowski University of Warsaw |
Informatyka Teoretyczna Graphs of bounded twin-width are quasi-polynomially χ-bounded |
We prove that for every t∈ℕ there is a constant γ(t) such that every graph with twin-width at most t and clique number ω has chromatic number bounded by 2γ(t) log^{4t+3} ω. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially χ-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially χ-bounded. This is a joint work with Michał Pilipczuk |
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30.04.46161 Ignacy Buczek |
Podstawy Informatyki Dömösi, Horváth and Ito’s Hypothesis on the Language of Primitive Words |
| A word is called primitive if it is not a repetition of another word. The language of all primitive words over a fixed alphabet \Sigma is denoted as Q. We consider the question of whether Q over \Sigma with at least 2 different characters is context-free or not. We show that Q is not regular and that it is context-sensitive. We exclude Q from language classes that are in-between the classes of regular languages and context-free languages, such as unambiguous context-free languages, linear context-free languages, and deterministic context-free languages. We also show that Q satisfies a number of context-free-like conditions, such as the Bar-Hillel lemma, the Ogden condition, the nonempty version of the strong Bader-Moura condition, and strengthened interchange property. In addition, we analyze some less typical (and unsuccessful) attempts of proving non-context-freeness of Q. |
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01.07.27105 Pat Morin Carleton University |
Informatyka Teoretyczna Free Sets in Planar Graphs |
A k-vertex set S of vertices in a planar graph G is free if, for any k-point set X, there exists a non-crossing straight-line drawing of G with the vertices of S mapped to the points in X. In this talk we survey the history and applications of free sets and present two recent results [1,2]: 1. Free sets and collinear sets: If G has any drawing in which all vertices of S appear on a line, then S is a free set. 2. Free sets and dual circumference: If G has bounded degree, then the size of the largest collinear set in G is proportional to the length of the longest cycle in the dual of G. [1] Vida Dujmović, Fabrizio Frati, Daniel Gonçalves, Pat Morin, and Günter Rote: Every collinear set in a planar graph is free. Discrete & Computational Geometry, 65:999–1027, 2021. [2] Vida Dujmovic, Pat Morin: Dual Circumference and Collinear Sets. SoCG 2019: 29:1-29:17. |
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24.02.7940 Jarosław Byrka University of Wrocław |
Informatyka Teoretyczna Online Facility Location with Linear Delay |
We study the problem of online facility location with delay. In this problem, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a penalty: each client incurs a waiting cost (the difference between its arrival and connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. This is a well-studied problem both of its own, and within the general class of network design problems with delays. Joint work with Marcin Bienkowski, Martin Böhm and Jan Marcinkowski |
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23.02.76419 Bartosz Podkanowicz |
Optymalizacja Kombinatoryczna Alon Tarsi number of planar graphs |
We prove that the Alon-Tarsi number of a planar graph is less or equal to 5. Alon Tarsi number is an important parameter for the graph. It is greater than the choice number and paintability for every graph. We show the modification of the standard argument presented by Carsten Thomassen. We construct a special orientation that doesn't have Euler subgraphs and allows us to reason about the Alon-Tarsi number. |
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16.11.76395 Jędrzej Kula |
Optymalizacja Kombinatoryczna Bipartite Perfect Matching is in quasi-NC |
The class NC represents the problems that have efficient parallel algorithms. In this work, the authors present two algorithms. The first algorithm proves that the perfect matching problem for bipartite graphs is in quasi-NC2. The second algorithm proves that the same problem is in the RNC class and uses only O(log2 n) random bits. Note that a complete derandomization would be achieved when the number of random bits comes down to O(log n). |
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22.07.76383 Krzysztof Pióro |
Optymalizacja Kombinatoryczna Graph coloring game |
In the game coloring game two players are given graph and a set of k colors. Players take turns, coloring properly an uncolored vertex. The goal of the first player is to complete the coloring of the graph, while the other one tries to prevent him from achieving it. The game chromatic number of a graph is the minimum number of colors needed for the first player to win. In this presentation I will show bounds for the game chromatic number for some classes of graphs. |
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02.10.73649 Sławomir Lasota University of Warsaw |
Informatyka Teoretyczna Improved Ackermannian lower bound for the reachability problem of vector addition systems |
Vector addition systems, equivalent to Petri nets, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there exists a sequence of valid execution steps reaching a given final configuration, is the central algorithmic problem for this model. The complexity of the problem has remained, until recently, one of the hardest open questions in verification of concurrent systems. A first upper bound has been provided only in 2015 by Leroux and Schmitz, then refined by the same authors to Ackermannian upper bound in 2019. The exponential space lower bound, shown by Lipton already in 1976, remained the only known for over 40 years until a breakthrough non-elementary lower bound by Czerwiński et al in 2019. Finally, a matching Ackermannian lower bound announced in 2021 by Czerwiński and Orlikowski, and independently by Leroux, established the complexity of the problem.
I will present an improvement of the former construction, making it conceptually simpler and more direct and, on the way, improving the lower bound for vector addition systems in fixed dimension (or, equivalently, Petri nets with fixed number of places). |
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28.03.73540 Jakub Fedak |
Podstawy Informatyki Exact enumeration of satisfiable 2-SAT formulae by Sergey Dovgal, Elie de Panafeu and Vlady Ravelomanana |
We obtain exact expressions counting the satisfiable 2-SAT formulae and describe the structure of associated implication digraphs. Our approach is based on generating function manipulations. To reject the combinatorial specificities of the implication digraphs, we introduce a new kind of generating function, the Implication generating function, inspired by the Graphic generating function used in digraph enumeration. Using the underlying recurrences, we make accurate numerical predictions of the phase transition curve of the 2-SAT problem inside the critical window. We expect these exact formulae to be amenable to rigorous asymptotic analysis using complex analytic tools, leading to a more detailed picture of the 2-SAT phase transition in the future.
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16.03.57218 Szymon Salabura |
Optymalizacja Kombinatoryczna The Hats game. On max degree and diameter |
In the Hats game, the sages, located at graph vertices, try to guess colors of their own hats, only seeing colors of hats on sages at the adjacent vertices. If using a deterministic strategy, at least one sage can guess the color of his own hat correctly, we say that the sages win. In this presentation, we consider the hat guessing number - the maximum number of possible colors, for which the sages can guarantee the win. We will see examples of graphs contradicting the previously stated hypothesis, that the hat guessing number is smaller than the graph's maximal degree + 1. We also show its independence from the graph's diameter. |
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27.05.54484 James Davies University of Waterloo |
Informatyka Teoretyczna Ringel's circle problem |
A constellation is a finite collection of circles in the plane in which no three circles are tangent at the same point. In 1959, Ringel asked how many colours are required to colour the circles of a constellation so that tangent circles receive different colours. We present a solution to Ringel's circle problem by constructing constellations that require arbitrarily many colours. Joint work with Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak |
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21.11.54374 Daniel Barczyk |
Podstawy Informatyki Narrow Proofs May Be Maximally Long by Albert Atserias and Massimo Lauria |
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size $n^{\Omega (w)}$. This shows that the simple counting argument that any formula refutable in width w must have a proof in size $n^{O (w)}$ is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.
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05.03.38065 Jacek Salata |
Optymalizacja Kombinatoryczna Choosability of K5-minor-free graphs |
Thomassen showed in 1994 that all planar graphs are 5-choosable and Škrekovski showed in 1998 that all K5-minor-free graphs also are 5-choosable. In this presentation we will take a look at two different proofs of the latter theorem: the Škrekovski's one from the original paper, and the one proposed by Wenjie Hea, Wenjing Miao and Yufa Shenb in 2007. |
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08.11.38052 Demian Banakh |
Optymalizacja Kombinatoryczna A relative of Hadwigers conjecture |
The well-known open Hadwiger's conjecture asserts that every simple graph G which is not t-colorable has Kt+1 minor. In this presentation, we will take a look at the proof of a relaxed version of this conjecture (in terms of so-called "defective colorings" - i.e. allowing a "small" number of monochromatic edges), as well as see how it can be useful for solving some other graph problems. |
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21.01.35319 Grzegorz Gutowski |
Informatyka Teoretyczna On a problem of Steinhaus |
In this talk, inspired by a "17-points" problem of Steinhaus (Problems 6 and 7 from his book "Sto zadań"), we discuss infinite sequences of real numbers in [0,1). |
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16.07.35209 Filip Synowiec |
Podstawy Informatyki On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface by Albert Atserias, Stephan Kreutzer and Marc Noy |
We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law – and not even a convergence law – on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.
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24.10.46278 Wojciech Buczek |
Optymalizacja Kombinatoryczna Parking functions: From combinatorics to probability |
Let's say m drivers have their favourite parking spot in the linear car park with n spots. Now, in some order, drivers will try to park their car at their favourite spot, and if they fail (because other car is standing there), they will try to park at the next avaible spot. If all drivers could park their car, we call this choices a parking function. In this seminar, we will look at this function proporties, create bijection from them to spanning forests and talk about some conjectures related to parking functions. |
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29.06.46266 Rafał Kilar |
Optymalizacja Kombinatoryczna A first moment proof of the Johansson-Molloy Theorem |
The paper provides a simple proof of a stronger version of Johansson-Molloy theorem, providing a bound on the list chromatic number of a graph based on maximum degree and neighbouhood density. The new proof only makes use of the first moment method. The counting argument used in the proof is inspired by work by Rosenfeld in the contex of non-repetitive graph coloring. The result is than extended to graphs with neighbourhoods with bounded density, which strengthens previous results. Lastly, the method is adapted to show asymptotically tight lowe bound on the number of colourings of sparse graphs . |
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10.09.43532 Lars Rohwedder EPFL |
Informatyka Teoretyczna A (2+ϵ)-approximation algorithm for preemptive weighted flow time on a single machine |
In a recent breakthrough in scheduling, Batra, Garg, and Kumar gave the first constant approximation algorithm for minimizing the sum of weighted flow times. Wiese and I [STOC'21] managed to improve this large unspecified constant to (2+ϵ). I will give a very graphic presentation of the algorithmic techniques behind this. |
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19.06.27113 Marcin Serwin |
Optymalizacja Kombinatoryczna Bears with Hats and Independence Polynomials |
A hat guessing game consists of a graph and bears assigned to vertices with a certain hat color. Each bear knows the colors of the bears belonging to the neighborhood of their vertex but does not know their own color. The bears win if at least one of them can guess the color of their hat. This presentation will introduce the aforementioned game, its variants and present findings of Václav Blažej, Pavel Dvořák and Michal Opler regarding fractional hat chromatic number of graphs with independence polynomials. |
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22.02.27101 Krzysztof Potępa |
Optymalizacja Kombinatoryczna Weak degeneracy of graphs |
The paper introduces a new graph parameter called "weak degeneracy", a variant of the degeneracy parameter. The authors show various applications of weak degeneracy. For example, it turns out that this new parameter is strongly correlated with graph coloring. Authors derive alternative proofs for several classic upper bounds in graph coloring theory, including 5-list-coloring of planar graphs. My presentation will summarize the findings of the paper. |
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06.05.24367 István Tomon ETH Zürich |
Informatyka Teoretyczna Ramsey properties of semilinear graphs |
A graph G is semilinear of complexity t if the vertices of G are points in some real space, and the edges of G are determined by the sign-patterns of t linear functions. Many natural geometric graph families are semilinear of bounded complexity, e.g. intersection graphs of boxes, shift graphs, interval overlap graphs. There is a long line of research studying the exceptional Ramsey and coloring properties of such geometric graphs; I will show that many of these results can be traced back to their semilinear nature. |
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29.10.24257 Katarzyna Król |
Podstawy Informatyki 0-1 Laws for Maps by Edward A. Bender, Kevin J. Compton,and L. Bruce Richmond |
A class of fnite structures has a 0-1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0-1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given,
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12.02.7948 Krzysztof Michalik |
Optymalizacja Kombinatoryczna Improved lower bound for the list chromatic number of graphs with no Kt minor |
This paper begins with recounting known limits regarding Hadwiger's conjecture and related problems including list Hadwiger's conjecture, stating that there does exist constant c, such that Kt minor free graph G has list coloring number not exceeding ct. After the introduction, we are presented with proof that such constant has to be at least equal to 2, contrary to previous results, where c was bounded by 4/3 and conjectured to be equal to 3/2. |
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18.10.7935 Krzysztof Ziobro |
Optymalizacja Kombinatoryczna Polynomials over structured grids |
Paper discusses properties of multivariate polynomials over finite grids, focaausing on he grids that are in some way "structured". To capture the degree to which a grid is structured, author introduces a notion of nullity, which can give us a numerical measure of structure. It is noted that for more structured grids we can obtain stronger versions of general theorems. This leads to the main results of the paper: the Structured Combinatorial Nullstellensatz and the Complete Coefficient Theorem. |
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29.12.5201 Barnaby Martin Durham University |
Informatyka Teoretyczna QCSP monsters and the future of the Chen Conjecture |
We elaborate the complexity of the Quantified Constraint Satisfaction Problem, QCSP(A), where A is a finite idempotent algebra. Such a problem is either in NP or is co-NP-hard, and the borderline is given precisely according to whether A enjoys the polynomially-generated powers (PGP) property. This completes the complexity classification problem for QCSPs modulo that co-NP-hard cases might have complexity rising up to Pspace-complete. Our result requires infinite languages, but in this realm represents the proof of a slightly weaker form of a conjecture for QCSP complexity made by Hubie Chen in 2012. The result relies heavily on the algebraic dichotomy between PGP and exponentially-generated powers (EGP), proved by Dmitriy Zhuk in 2015, married carefully to previous work of Chen. Finally, we discuss some recent work with Zhuk in which the aforementioned Chen Conjecture is actually shown to be false. Indeed, the complexity landscape for QCSP(B), where B is a finite constraint language, is much richer than was previously thought. |
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23.06.5092 Ignacy Buczek |
Podstawy Informatyki Definability on a Random 3-CNF Formula by Albert Atserias |
We consider the question of certifying unsatisfiability of random 3-CNF formulas. At which densities can we hope for a simple sufficient condition for unsatisfiability that holds almost surely? We study this question from the point of view of definability theory. The main result is that first-order logic cannot express any sufficient condition that holds almost surely on random 3-CNF formulas with $n^{2-\alpha}$ clauses, for any irrational positive number \alpha. In contrast, it can when the number of clauses is $n^{2+\alpha}$, for any positive \alpha. As an intermediate step, our proof exploits the planted distribution for 3-CNF formulas in a new technical way. Moreover, the proof requires us to extend the methods of Shelah and Spencer for proving the zero-one law for sparse random graphs to arbitrary relational languages.
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23.11.70919 Artur Salawa |
Optymalizacja Kombinatoryczna The Open Problems Project |
Paper records open problems in computational geometry and related fields. For every problem, the authors provide a statement, origin, current status, partial results and related problems. My presentation focuses on a few chosen problems explained in a friendly manner. |
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29.07.70907 Grzegorz Wawrzesta |
Optymalizacja Kombinatoryczna Density conditions for panchromatic colourings of hypergraphs |
A hypergraph is defined as a pair H = (V, E), where V is a set of vertices and E is a set of subsets of V - these subsets of vertices are called (hyper)edges. Graphs can be then seen as a concretization where all edges are sets of size 2. This can be shortly ascribed to the hypergraph as being 2-uniform. T-uniformity is a useful assumption for deriving its properties but sometimes one would wish for more general results. This approach is one of a few that are considered by the authors of the following paper which focuses on boundaries we can put on some characteristic properties of hypergraphs relating to their colorability and list-colorability. During the meeting, the basic concepts of hypergraphs and their colorability will be introduced and then the results of the paper will be interpreted alongside the presentation of the theorems and lemmas (and also an exemplar proof of one of them or two) which are used in the paper to attain the results. |
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02.12.70856 Miłosz Januszewski, Szymon Salabura |
A Fine-Grained Perspective on Approximating Subset Sum and Partition |
W problemie Subset Sum, mając dany zbiór dodatnich liczb X oraz cel t, pytamy czy istnieje dowolny podzbiór sumujący się do dokładnie t. Należy on do klasy problemów NP-zupełnych, zatem naturalne jest badanie jego algorytmów aproksymacyjnych - takich, które szukają podzbioru sumującego się do co najmniej 1-ε wyniku optymalnego. Aktualnie najlepszy znany algorytm robi to w czasie O(min{n/ε,n+1/ε2 log(1/ε)}). W szczególności nie jest znany żaden algorytm rozwiązujący ten problem w O((n+1/ε)c) dla dowolnego c<2. Autorzy w pracy pokazują równoważność tego problemu do Min-Plus-Convolution przeprowadzając redukcje w obie strony. Dzięki temu uzyskują algorytm aproksymacyjny dla Subset Sum o poprawionym czasie działania oraz udowadniają, że Subset Sum ma algorytm aproksymacyjny w czasie O((n+1/ε)c) dla c<2 wtedy i tylko wtedy, gdy Min-Plus-Convolution może być rozwiązany w O(nc') dla c'<2. Druga część równoważności jest jednak sprzeczna z hipotezą trudności tego problemu. Dodatkowo, dla specjalnego wariantu Subset Sum zwanego Partition, autorzy stosują powyższą redukcję otrzymując algorytm aproksymacyjny działający w czasie O(n+1/ε3/2). Jest to pierwszy deterministyczny algorytm z podkwadratową złożonością. |
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09.10.68173 Jan Derbisz |
Informatyka Teoretyczna Circular-arc graphs and the Helly property |
Circular-arc graphs, defined as the intersection graphs of arcs of a fixed circle, are probably one of the simplest classes of intersection graphs, which does not satisfy the Helly property in general (i.e. there are circular-arc graphs in which some cliques can be represented by arcs whose common intersection is empty). In particular, some cliques of a circular-arc G graph may satisfy the Helly property in some but not all circular-arc representations of G. In the Helly Clique Problem, for a given circular-arc graph G and some of its cliques C1,...,Ck (not necessarily maximal in G) one needs to decide whether there exists a circular-arc representation of G in which all the cliques C1,...,Ck satisfy the Helly property. We know that the Helly Clique Problem is NP-complete and, under the ETH, it can not be solved in time 2o(k)poly(n), where n is the number of vertices of G (Ağaoğlu et al.). In the talk we will consider the Helly Clique Problem in the context of parameterized complexity, where the natural parameter is the number of cliques given in the input. We will show that the Helly Clique Problem: Moreover, we will discuss the relation of the Helly Clique Problem with the recognition problems of so-called H-graphs; in particular, in the context of those graphs H for which the recognition problem remains open. This is joint work with T. Krawczyk. The talk also includes joint work with D. Ağaoğlu, O. Cagrici, T. Hartmann, P. Hliněný, J. Kratochvíl, T. Krawczyk, and P. Zeman. |
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03.04.68064 Michał Woźny |
Podstawy Informatyki Dance of the Starlings by Henk Barendregt, Jorg Endrullis, Jan Willem Klop, and Johannes Waldmann |
In this birdwatching paper our binoculars are focused upon a particular bird from Smullyan's enchanted forest of combinatory birds, to wit the Starling. In the feathers of lambda-calculus this bird has the plumage \abc:ac(bc). This term is usually named S, reminiscent of its inventor Schonfinkel and also the combinatory ornithologist Smullyan. The combinator S is important for a variety of reasons. First, it is part of the S, K -basis for Combinatory Logic (CL). Second, there are several interesting questions and observations around S, mostly referring to termination and word problems. Our paper collects known facts, but poses in addition several new questions. For some of these we provide solutions, but several tough open questions remain.
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18.07.51754 Maciej Nemś |
Optymalizacja Kombinatoryczna Fair Correlation Clustering |
In this paper authors propose approximation for Correlation Clustering problem with additional constaint of fairness. In a fair version of correlation clustering vertices have also colors and in the end in each cluster there should be "some" number of vertices of each color. What "some" means is dependent on variant of this constraint. Authors first reduce a problem of fair clustering correlation to a problem they call Fairlet Decomposition and then show approximation algorithm for this problem. In the end they describe some experiments they have done to prove Fair Correlation Clustering a viable version of Correlation Clustering. |
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23.03.51742 Karolina Gontarek |
Optymalizacja Kombinatoryczna Growth properties of power-free languages |
Paper surveys common part of two formal language issues. Issue of repetition free words and languages and issue of growth functions for words and languages. Paper gives an overview of current knowledge and search about an intersection of those two areas. It classifies power-free languages with regard to their growth rate. It also describes methods of esimating complexity of power-free languages paying attention to amount of computer resources needed by special method. Finally, it presents future directions of research in this area. |
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28.07.51691 Aleksander Katan, Roch Wójtowicz |
Filling crosswords is very hard |
Autorzy analizują problem wypełniania krzyżówek, który już był rozważany na przykład przez Garey’a i Johnsona w ich książce „Computers and Intractability: A Guide to the Theory of NP-Completeness”. W problemie tym dostajemy m słów i n poziomych lub pionowych slotów (rubryk) oraz jesteśmy proszeni o wypełnienie ich tak, by przecięcia slotów się zgadzały. Autorzy próbują wskazać źródło trudności tej łamigłówki przyglądając się strukturze grafu przecięć slotów. Skupiają się na przypadku, gdy struktura tego grafu przypomina drzewo. Niestety, jeżeli przyjmiemy, że słowa nie mogą być używane wielokrotnie, okazuje się, że problem pozostaje NP-trudny nawet pod bardzo surowymi restrykcjami, jak na przykład, że graf przecięć jest sumą grafów typu star i alfabet ma rozmiar |
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04.06.49008 Nicolas Bousquet CNRS, Lyon |
Informatyka Teoretyczna Local certification of/on sparse graph classes |
Local certification consists in assigning labels to the nodes of a graph in order to certify that some given property is satisfied, in such a way that the labels can be checked locally. In this talk, our goal is to certify that a graph G belongs to a given graph class. Such certifications exist for many sparse graph classes such as trees, planar graphs and graphs embedded on surfaces with labels of logarithmic size. It is open if such a certificate exist for any H-minor free graph class. We present some generic tools which allow us to certify the H-minor-free graphs (with logarithmic labels) for each small enough H. More generally, we consider classes defined by any MSO formula (i.e. the MSO-model checking problem), and show a local version of the well-known Courcelle theorem: in bounded treedepth graphs, logarithmic certificates can be obtained for any MSO formula. We will also discuss many open problems related to local certification of/on sparse graph classes. Joint works with Laurent Feuilloley and Théo Pierron |
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27.11.48898 Łukasz Selwa |
Podstawy Informatyki An Inverse of the Evaluation Functional for Typed λ-calculus by U. Berger and Η. Schwichtenberg |
| In any model of typed lambda-calculus containing some basic arithmetic, a functional p ->e (procedure —> expression) will be defined which inverts the evaluation functional for typed lambda-terms. Combined with the evaluation functional, p->e yields an efficient normalization algorithm. The method is extended to lambda-calculi with constants and is used to normalize (the lambda-representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for \beta \eta-reduction, generalizing results of Friedman and Statman. If two lambda-terms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are provably equal in the \beta \eta-calculus. |
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27.11.48898 Łukasz Selwa |
An Inverse of the Evaluation Functional for Typed λ-calculus by U. Berger and Η. Schwichtenberg |
| In any model of typed lambda-calculus containing some basic arithmetic, a functional p ->e (procedure —> expression) will be defined which inverts the evaluation functional for typed lambda-terms. Combined with the evaluation functional, p->e yields an efficient normalization algorithm. The method is extended to lambda-calculi with constants and is used to normalize (the lambda-representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for \beta \eta-reduction, generalizing results of Friedman and Statman. If two lambda-terms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are provably equal in the \beta \eta-calculus. |
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28.01.29843 Zdeněk Dvořák Charles University |
Informatyka Teoretyczna On asymptotic dimension of planar and geometric graphs |
A graph class C has asymptotic dimension at most d if for every r, the r-th distance power of any graph from C can be colored by d+1 colors so that every monchromatic connected subgraph has bounded weak diameter. In a recent breakthrough result, Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott proved that all proper minor-closed classes have asymptotic dimension at most two. We investigate some questions motivated by this result for planar graphs and geometric intersection graphs. Joint work with Sergey Norin |
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06.11.13423 Roch Wójtowicz |
Optymalizacja Kombinatoryczna Problems and results on 3-chromatic hypergraphs and some related questions |
Authors in this work aim to establish various bounds and constraints on hypergraphs which are k-chromatic. Hypergraph is a graph where an edge don’t have to link exactly two vertices. Hypergraph is called simple, when none two of his edges has more then one common point, and is called clique when each two of his edges has at least one common point. Hyper graph is r-uniform when each of its edges contains exactly r points. Chromatic number is a smallest number k such that you can color points of the graph using k colors in the way that no edge is monochromatic. Main part of the work involves around the impact that being clique or simple has on 3-chromatic hypergraph structure. The main reason why those two things are connected is following trivial observation: If a hypergraph has chromatic number > 3 then it has two edges with exactly one common point.
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12.07.13411 Grzegorz Gawryał |
Optymalizacja Kombinatoryczna Defective and clustered choosability of sparse graphs |
This paper explores almost proper graph colorings and list colorings - we allow the coloring to be improper, but we impose restrictions on the maximum number of neighbours of any vertex with the same color as the vertex itself (defect) or the maximum allowed size of a monochromatic connected graph component (clustering). The paper provides new bounds on coloring and list coloring number for sparse graphs, i.e. having bounded maximum average degree, taken over all subgraphs, or limited maximum degree. More precisely, the two main results of this paper are the new bounds on defective choosability and clustered choosability of graphs with bounded maximum average degree, being the best known results for graphs with unbounded maximum degree, but bounded maximum average degree, like k-stack and k-queue graphs. |
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15.11.13360 Mateusz Pach, Michał Wronka |
Determining 4-edge-connected components in linear time |
| Prezentujemy pierwszy deterministyczny algorytm obliczający 4-spójne krawędziowo składowe w czasie liniowym. Najpierw pokazujemy algorytm znajdujący wszystkie 3-cięcia krawędziowe w danym grafie 3-spójnym krawędziowo i korzystając z jego wyniku budujemy 4-spójne składowe oryginalnego grafu. |
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22.09.10677 Torsten Mütze University of Warwick & Charles University |
Informatyka Teoretyczna Efficient generation of elimination trees and Hamilton paths on graph associahedra |
An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G−x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they are closely related to centered colorings and tree-depth. They also encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a simple greedy algorithm (see www.combos.org/elim). This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time O(m+n) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time O(1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of G if and only if G is chordal.
This is joint work with Jean Cardinal (ULB) and Arturo Merino (TU Berlin) |
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17.03.10568 Juliusz Wajgelt |
Podstawy Informatyki On Repetitive Right Application of B-Terms by Mirai Ikebuchi and Keisuke Nakano |
| B-terms are built from the B combinator alone defined by B \f.\g.\x.f (g x), which is well known as a function composition operator. This paper investigates an interesting property of B-terms, that is, whether repetitive right applications of a B-term cycles or not. We discuss conditions for B-terms to have and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of B-terms which have the property and show that there are infinitely many B-terms which do not have the property. Also, we introduce a canonical representation of B-terms that is useful to detect cycles, or equivalently, to prove the property, with an efficient algorithm. |
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23.07.79070 Piotr Kaliciak, Kamil Galewski |
Turing Completeness and Sid Meier’s Civilization |
| W pracy zostało wykazane, że trzy gry strategiczne z serii Sid Meier's Civilization: Sid Meier’s Civilization: Beyond Earth, Sid Meier’s Civilization V, i Sid Meier’s Civilization VI, są zupełne w sensie Turinga. Dla każdej gry została pokazana, oparta na jej mechanikach, konstrukcja uniwersalnej maszyny Turinga. Istnienie takich maszyn oznacza, że pod pewnymi założeniami gry te są nierozstrzygalne. Praca pokazuje działanie przykładowej maszyny - Zajętego Bobra złożonego z trzech stanów, zaimplementowanej w jednej z gier. |
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22.11.76277 Jan Kościsz |
Podstawy Informatyki FIXED POINT COMBINATORS AS FIXED POINTS OF HIGHER-ORDER FIXED POINT GENERATORS by ANDREW POLONSKY |
| Corrado Bohm once observed that if Y is any fixed point combinator (fpc), then Y (\yx:x(yx)) is again fpc. He thus discovered the first \fpc generating scheme" a generic way to build new fpcs from old. Continuing this idea, define an fpc generator to be any sequence of terms G_1, ..., G_n such that Y is fpc then Y G_1...G_n is fpc: In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several robust classes of such generators, by examining their elementary properties like injectivity and (weak) constancy. We provide sufficient conditions for existence of fixed points of a given generator (G_1, ..., G_n): an fpc Y such that Y = Y G_1 ... G_n. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This statement generalizes Statman's conjecture on non-existence of "double fpcs": fixed points of the generator (G) = (\yx:x(yx)) discovered by Bohm. Finally, we define and make a few observations about the monoid of (weak) fpc generators. This enables us to formulate new conjectures about their structure. |
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02.07.73508 David Wood Monash University |
Informatyka Teoretyczna Universality in minor-closed graph classes* |
Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain an infinite complete graph as a minor. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has O(n1/2) treewidth (which implies the Lipton-Tarjan separator theorem). More generally, for every fixed positive integer t we construct a countable graph that contains every countable Kt-minor-free graph and has the above key properties. Joint work with Tony Huynh, Bojan Mohar, Robert Šámal and Carsten Thomassen * exceptionally: Tuesday at 11:00 |
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18.03.59905 Daniel Bobrzyk, Mateusz Golonka |
Wake Up and Join Me! An Energy-Efficient Algorithm for Maximal Matching in Radio Networks |
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22.01.57222 Bartłomiej Kielak |
Informatyka Teoretyczna Inducibility of small oriented graphs |
For a fixed graph H, let i(H, n) be the maximum induced density of H in any graph on n vertices. The limit i(H, n), as n goes to infinity, always exists and is called inducibility of H. Fox, Huang, and Lee proved that for almost all graphs H (think of large 'typical' graphs), inducibility of H can be obtained as the limit of induced density of H in its iterated blowups. Apart from that, inducibility is well understood only for narrow classes of graphs; in particular, it is still not determined for H being a path on 4 vertices. Definition of inducibility can be easily adapted to different settings of combinatorial structures. In this talk, I will focus on the setting of oriented graphs and discuss the inducibility of oriented graphs on at most 4 vertices.
Joint work with Łukasz Bożyk and Andrzej Grzesik |
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06.07.40790 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Reconfiguring Independent Sets on Interval Graphs |
In the reconfiguration problem, we are given a set of objects and rules of how one object can be reconfigured into another one. The main questions to be asked are if it is possible to reconfigure two given objects into each other (Reachability Problem) or how long is the shortest possible reconfiguration sequence. We focus on reconfiguring independent sets in a given graph. Two independent sets are reconfiguration-adjacent if their symmetric difference consists exactly of two vertices connected by an edge. It is known that for some graph classes the Reachability Problem can be solved in polynomial time. I briefly survey the topic and show that the problem is computationally hard for incomparability graphs. Moreover, I discuss the reconfiguration paths length problem in general and in more detail in the class of interval graphs. |
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16.09.38056 Daniel Kráľ Masaryk University in Brno |
Informatyka Teoretyczna Uniform Turán density of hypergraphs |
In the early 1980s, Erdős and Sós, initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K43, the complete 4-vertex 3-uniform hypergraph, and K43-, the hypergraph K43 with an edge removed. The latter question was solved only recently in [Israel J. Math. 211 (2016), 349–366] and [J. Eur. Math. Soc. 97 (2018), 77–97], while the former still remains open for almost 40 years. Prior to our work, the hypergraph K43- was the only 3-uniform hypergraph with non-zero uniform Turán density determined exactly. During the talk, we will present the following two results:
The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper, Frederik Garbe, Ander Lamaison, Samuel Mohr and David Munhá Correia. |
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08.04.21629 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Open problem session |
Several open problems related to 1-2-3 Conjecture are presented. |
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12.05.18891 Virginia Vassilevska Williams MIT |
Informatyka Teoretyczna A refined laser method and (slightly) faster matrix multiplication |
Matrix multiplication is one of the most basic linear algebraic operations outside elementary arithmetic. The study of matrix multiplication algorithms is very well motivated from practice, as the applications are plentiful. Matrix multiplication is also of great mathematical interest. Since Strassen's discovery in 1969 that n-by-n matrices can be multiplied asymptotically much faster than the brute-force O(n3) time algorithm, many fascinating techniques have been developed, incorporating ideas from computer science, combinatorics, and algebraic geometry. The fastest algorithms over the last three decades have used Strassen's "laser method" and its optimization by Coppersmith and Winograd. The method has remained unchanged; the algorithms have differed in what the method was applied to. In recent work, joint with Josh Alman, we improve the method so that applying it to the same objects that the old method was applied to immediately yields faster algorithms. Using this new method, we obtain the theoretically fastest algorithm for matrix multiplication to date, with running time O(n2.37286). This talk will give an overview of the main techniques and will also outline our recent improvement of the laser method. |
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12.04.49031 Szymon Salabura |
Optymalizacja Kombinatoryczna The Fixing Block Method in Combinatorics on Words |
A word is repetitive if it contains two consecutive identical blocks. A sequence is non-repetitive up to mod r if each of its mod k (1⩽k⩽r) subsequences is non-repetitive. Let L be a language of non-repetitive (up to mod r) words. In this seminar, we are going to take a look at fixing blocks - a special kind of suffixes preventing words of L to have an extension in L. Using the fixing blocks method we are going to show some interesting properties of such languages. We also outline a method of attack for more complex problems.
(the seminar will only be online) |
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03.01.49008 Wojciech Węgrzynek |
Optymalizacja Kombinatoryczna Non-repetetive words: ages and essences |
The age of an infinite word will be the set of all its finite subwords, it's essence will be the set of all finite subwords occurring infinitely many times. The language L{121,323} is the language of all square-free infinite words, such that they don’t have 121 or 323 as subwords. It turns out if we consider the equivalence relations on L{121,323} in respect to the ages and the essences we will get an uncountable cardinality of equivalence classes and 1 equivalence class respectively.
(the seminar will only be online) |
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05.02.46270 Krzysztof Turowski |
Informatyka Teoretyczna Degree Distribution of Dynamic Graphs Generated by a Duplication-Divergence Models |
We analyze the structure of dynamic graphs generated from duplication models in which a new vertex selects an existing vertex and copies some of its neighbors and then we add some random divergence. We analyze various graph parameters like mean degree, number of open triangles, number of triangles, number of vertices of degree k or maximum degree in a graph generated from such models. We provide asymptotic analysis of expected values and tail behavior of these parameters. We also point to further extensions of this approach towards computing symmetries in these graphs and algorithms for graph compression.
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05.12.29865 Bartosz Wodziński |
Optymalizacja Kombinatoryczna Zarankiewicz's Problem and some related results |
In 1951, Kazimierz Zarankiewicz posed a problem asking for the largest possible number of edges in a bipartite graph that has a given number of vertices (n) and has no complete bipartite subgraphs of a given size. Although solved for some specific cases, it still remains open in general. It led to some interesting results in extremal graph theory, such as Kővári–Sós–Turán theorem which gives an upper bound for this problem. During the seminar, I will discuss several problems related to forbidding subgraphs, from easy up to unsolved ones. I will also show their connection with some geometric problems such as creating a maximum number of unit distances between n points on a plane.
(the seminar will only be online) |
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28.08.29842 Michał Zwonek |
Optymalizacja Kombinatoryczna Polyomino Tilings |
A polyomino is a subset of R2 formed by a strongly connected union of axis-aligned unit squares centered at locations on the square lattice Z2. Let T = {T1,T2,...} be an infinite set of finite simply connected closed sets of R2. Provided the elements of T have pairwise disjoint interiors and cover the Euclidean plane, then T is a tiling and the elements of T are called tiles. Provided every Ti ∈ T is congruent to a common shape T, then T is monohedral, T is the prototile of T, and the elements of T are called copies of T. In this case, T is said to have a tiling. We will go through some of the open problems related to polyomino tilings. (the seminar will only be online) |
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01.10.27104 Paweł Rzążewski Warsaw University of Technology |
Informatyka Teoretyczna Treewidth of graphs with forbidden induced subgraphs |
The notion of treewidth and tree decompositions plays a central role in the study of graphs with forbidden minors. Besides structural characterizations, the property of having boundedtreewidth, or a tree decomposision with certain "nice" properties, can be used algorithmically. However, until very recently, there were very few results that allowed to analyze the treewidth of graphs that exclude certain induced subgraphs. Indeed, a large clique has large treewidth, but is H-free for any graph H which is not a clique. It turns out that some intresting relations between the two worlds can be found if we additionally put some restrictions on vertex degrees - either just by bounding the maximum degree, or, in some cases, by bounding the degeneracy. During the talk we will discuss some results of this flavor. In particular, we will show that
Based on the joint work with Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and with Abrishami, Chudnovsky, and Dibek. |
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30.01.24171 Bartosz Walczak |
Informatyka Teoretyczna Coloring polygon visibility graphs and their generalizations |
The visibility graph of a polygon P is formed by the pairs of vertices u and v of P such that the segment uv is disjoint from the exterior of P. We show that the class of polygon visibility graphs is χ-bounded, thus answering a question by Kára, Pór, and Wood from 2005. Specifically, we prove the bound χ≤3⋅4ω−1. We obtain the same bound for generalizations of polygon visibility graphs in which the polygon is replaced by a curve and straight-line segments are replaced by segments in a pseudo-line arrangement. The proof is carried through in the setting of ordered graphs. In particular, we show χ-boundedness of several classes of ordered graphs with excluded ordered substructures. Joint work with James Davies, Tomasz Krawczyk, and Rose McCarty. This is a part of Round the World Relay in Combinatorics organized by Oxford University. Here is the full schedule: http://people.maths.ox.ac.uk/scott/relay.htm And the zoom link for the whole event: |
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27.05.7939 Marthe Bonamy Université de Bordeaux |
Informatyka Teoretyczna Graph recolouring |
Given a solution to a problem, we can try and apply a series of elementary operations to it, making sure to remain in the solution space at every step. What kind of solutions can we reach this way? How fast? This is motivated by a variety of applications, from statistical physics to real-life scenarios, including enumeration and sampling. In this talk, we will discuss various positive and negative results, in the special case of graph colouring. |
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06.04.76410 Jan Mełech |
Optymalizacja Kombinatoryczna Rödl Nibble |
For positive integers r<k<n let m(n,k,r) be the maximal size of a family F of k-element subsets of [n] such that no r vertices lie in more than one A in F. The Erdös-Hanani conjecture states that as n grows to infinity m(n,k,r) tends to (n choose r)/(k choose r). Firstly, we will see a sketch of the proof of this conjecture proposed by Vojtech Rödll. Then we will talk about how this is connected with packing in hypergraph and discuss the idea of an algorithm called Rödl nibble that achieves asymptotically optimal packing k-uniform hypergraphs. (the seminar will only be online) |
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28.12.76386 Krzysztof Pióro |
Optymalizacja Kombinatoryczna Decomposing planar graphs into graphs with degree restrictions |
Given a graph G, its (d,h)-decomposition is a partition of a set of edges of this graph into a d-degenerate graph and a graph with maximum degree at most h. We will study (d,h)-decomposability of planar graphs and consider the problem of finding minimum hd such that every planar graph is (d,hd)-decomposable. Since every planar graph is 5-degenerate, we will consider only the case of d less than 5. (the seminar will only be online) |
Poprzednie referaty
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26.05.2021 Piotr Kawałek |
Informatyka Teoretyczna Constant depth circuits |
We will overview the state-of-the-art results and techniques used in proofs of the lower bounds for constant depth circuits. We focus mostly on classes AC[0], ACC[0] and CC[0]. The most important challenges and some open problems are to be discussed. |
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29.11.57244 Maciej Nemś |
Optymalizacja Kombinatoryczna Ant Colony Optimization |
Ant Colony Optimization algorithms are part of swarm intelligence approach to solving problems. They are inspired by behavior of ants. After finding a desired destination ants leave pheromones on the way back to the colony. This way all ants can detect the scent and decide to go in the direction suggested by pheromone trail. ACO is based on this concept and involves multi-agent computation. Communication between agents is done by changing the stimuli for all agents, to make a certain action. This is similar to ants leaving pheromones. Presentation will include basic concept of Ant Colony Optimization and an example of solving a well known problem using it. I will also present a formalization of ACO into a metaheuristic for combinatorial optimization problems. Presentation will end with talk about current state of ACO, its limitation and possible future.
(the seminar will only be online) |
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22.08.57221 Wojciech Buczek |
Optymalizacja Kombinatoryczna Woodall’s conjecture |
Woodall’s conjecture tells us, that any directed cut with at least k edges has at least k disjoint dijoins. Set of edges D is a dijoin if and only if the digraph (V, E ∪ D-1) is strongly connected. We will say about the linear programming formulation of this problem, equivalent and related problems to it, and some partial results by Shrijver, Lee and Wakabayashi, and Meszaros. We will also show counterexamples to a generalized version of the conjecture.
(the seminar will only be online) |
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25.09.54483 Paweł Idziak |
Informatyka Teoretyczna Modular circuits satisfiability of intermediate complexity |
In our paper [LICS'18] a generalization of Boolean circuits to arbitrary finite algebras was introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However nilpotent but not supernilpotent algebras have not been put on any side of this borderline. This paper provides a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce Boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general. Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy was shown by Bulatov and Zhuk. Joint work with Piotr Kawałek and Jacek Krzaczkowski |
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16.04.38056 Vladyslav Rachek, Ruslan Yevdokymov |
Optymalizacja Kombinatoryczna An Introduction to the Discharging Method via Graph Coloring |
The discharging method is a technique that can be used to show that some global properties of a graph imply the existence of local structures. Then, we can sometimes show, that such structures imply that the graph has another global property. The discharging method is thus a "connector" between global properties of a graph (via local properties, e.g. having subgraphs or minors). In the first part of the presentation, we talk about the structure and coloring of sparse and plane graphs. Typical statements will sound like "If there is some global degree bound, then the chromatic number is somehow bounded"
(the seminar will only be online) |
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21.05.35318 Grzegorz Gutowski |
Informatyka Teoretyczna Filling blanks in matrices to avoid singularity: progress report |
Given an n-by-n generator matrix with entries being subsets of a fixed field we generate the set of all matrices with entries coming from the corresponding entries in the generator matrix. Such a set of matrices is strongly singular if each member is a singular matrix. In this talk I will survey natural generalizations and connections to other problems. In particular, I will describe algorithm by Geelen for maximum rank matrix completion problem. |
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20.03.18914 Marcin Serwin |
Optymalizacja Kombinatoryczna Aanderaa-Karp-Rosenberg conjecture |
The conjecture deals with queries on graph. More concretely given property of a graph (such as connectedness or non-emptiness) we may ask whether it is possible to recognize a graph with this property without querying all of its edges. It turns out that for many properties it is indeed not possible to do so in a deterministic manner for all graphs. The Aanderaa–Karp–Rosenberg conjecture states that any non-trivial monotone graph property cannot be determined by a deterministic algorithm with less than n(n-1)/2 queries. Such graph properties are called evasive, thus this conjecture is sometimes called evasiveness conjecture. (the seminar will only be online) |
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10.12.18890 Krzysztof Potępa |
Optymalizacja Kombinatoryczna Orienting Fully Dynamic Graphs with Worst-Case Time Bounds |
In the edge orientation problem, one is asked to orient edges of a given graph such that the out-degrees of vertices are bounded by some function. In the fully dynamic variant, we want to process arbitrary edge insertions and deletions in an online fashion. We will show an algorithm for graphs with bounded arboricity that achieves logarithmic out-degree bound and supports updates in O(log n) worst-case time.
(the seminar will only be online) |
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13.01.16153 Louis Esperet Université Grenoble Alpes |
Informatyka Teoretyczna Universal graphs and labelling schemes |
Given a graph class C, a graph G is universal for C if it "contains" all the graphs from C. As there are several notions of containment, there are several notions of universal graphs. In this talk I'll mention two versions:
Note that an isometric copy is an induced copy, so the second notion is stronger. These notions are closely related to the notion of labelling schemes in graphs. The goal is to assign labels to the vertices of each graph G from C such that upon reading the labels of any two vertices u and v, we know some properties of u and v in G (whether they are adjacent, or their distance, or whether u is reachable from v if G is a digraph). It turns out that minimizing the size of the labels is equivalent to constructing small universal graphs, at least in the case of induced-universal graphs. For isometric-universal graphs some additional work needs to be done. I'll survey some recent progress in this area. In particular I'll show how to construct induced-universal graphs of almost optimal size for any hereditary class, using the regularity lemma. In particular this implies almost optimal reachabilty labelling schemes in digraphs and comparability labelling schemes in posets, and the construction of an almost optimal universal poset for the class of all n-element posets (of size 2n/4+o(n)). I will also show how to construct isometric-universal graphs of size 3n+o(n) for the class of all n-vertex graphs, answering a question of Peter Winkler. Based on joint work with Marthe Bonamy, Cyril Gavoille, Carla Groenland, and Alex Scott. |
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21.09.81862 Mateusz Kaczmarek |
Optymalizacja Kombinatoryczna On triangles in Kr-minor free graphs |
There is a close connection between minors of the graph and a lower bound on such number k that each edge (or vertex) belongs to at least k triangles. One of the most interesting classes of minors is the class of complete graphs Kr. In the paper 'On triangles in Kr-minor free graphs', Boris Albar and Daniel Gonçalves take a closer look at this class of graphs. Based on their work I will present some interesting results regarding this connection and show how it correlates with Hadwiger's conjecture.
(the seminar will only be online) |
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25.10.79124 Daniel Kráľ Masaryk University in Brno |
Informatyka Teoretyczna Quasirandom combinatorial structures |
A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. The notion of quasirandom graphs, developed in the work of Rödl, Thomason, Chung, Graham and Wilson in 1980s, is particularly robust as several different properties of truly random graphs, e.g., subgraph density, edge distribution and spectral properties, are satisfied by a large graph if and only if one of them is. We will discuss quasirandom properties of other combinatorial structures, tournaments, permutations and Latin squares in particular, and present several recent results obtained using analytic tools of the theory of combinatorial limits. The talk is based on results obtained with different groups of collaborators, including Timothy F. N. Chan, Jacob W. Cooper, Robert Hancock, Adam Kabela, Ander Lamaison, Taísa Martins, Roberto Parente, Samuel Mohr, Jonathan A. Noel, Yanitsa Pehova, Oleg Pikhurko, Maryam Sharifzadeh, Fiona Skerman and Jan Volec. |
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16.05.62697 Bartłomiej Jachowicz |
Optymalizacja Kombinatoryczna Acyclic coloring of graphs with fixed maximum degree |
An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted as a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. Known problem in this area is to find an upper bound for an acyclic chromatic number for graphs with a fixed maximum degree. One of the first papers on this topic is Hocquard's article "Graphs with maximum degree 6 are acyclically 11-colorable". The proofing technique from his work has been used in many later papers that show similar bounds for graphs with fixed maximum grades.
(the seminar will only be online) |
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20.06.59959 Paweł Gawrychowski University of Wrocław |
Informatyka Teoretyczna Fully Dynamic Longest Increasing Subsequence |
We revisit the problem of maintaining the longest increasing subsequence (LIS) of an array under (i) inserting an element, and (ii) deleting an element of an array. In a recent breakthrough, Mitzenmacher and Seddighin [STOC 2020] designed an algorithm that maintains an O((1/ϵ)O(1/ϵ))-approximation of LIS under both operations with worst-case update time ~O(nϵ), for any constant ϵ>0. We exponentially improve on their result by designing an algorithm that maintains a (1+ϵ)-approximation of LIS under both operations with worst-case update time ~O(ϵ−5). Instead of working with the grid packing technique introduced by Mitzenmacher and Seddighin, we take a different approach building on a new tool that might be of independent interest: LIS sparsification. While this essentially settles the complexity of the approximate version of the problem, the exact version seems more elusive. The only known sublinear solution was given very recently by Kociumaka and Seddighin [STOC 2021] and takes ~O(n2/3) time per update. We show polynomial conditional lower bounds for two natural extensions of this problem: weighted LIS and LIS in any subarray. Joint work with Wojciech Janczewski
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10.01.43532 Piotr Mikołajczyk |
Optymalizacja Kombinatoryczna Thomassen's choosability argument revisited |
The Hadwiger Conjecture states that if a graph G does not contain a clique on t vertices as a minor, then G is (t-1)-colorable. For low values of t (<7) it was already shown that this claim is actually true. Currently, the best-known upper bound on the chromatic number of Kt-minor-free graphs is O(ct*sqrt(log(t))) and the proof follows from a degeneracy argument. Unfortunately, this approach cannot be exploited further. However, by revisiting Thomassen's reasoning from '94 we can try to prepare the ground for a new way of attacking the Hadwiger Conjecture based on graph choosability. For that, we will take a look at a new proof of a theorem that every K5-minor-free graph is 5-choosable.
(the seminar will only be online) |
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12.02.40794 Michał Seweryn |
Informatyka Teoretyczna Dimension of posets with k-outerplanar cover graphs |
In 2015, Felsner, Trotter, and Wiechert showed that posets with outerplanar cover graphs have bounded dimension. We generalise this result to posets with k-outerplanar cover graphs. Namely, we show that posets with k-outerplanar cover graph have dimension O(k3). As a consequence, we show that every poset with a planar cover graph and height h has dimension O(h3). This improves the previously best known bound of O(h6) by Kozik, Micek and Trotter. Joint work with Maximilian Gorsky |
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04.09.24366 Jędrzej Kula |
Optymalizacja Kombinatoryczna Combinatorial Nullstellensatz |
Proposed by Noga Alon in 1999 an algebraic technique inspired by Hilbert’s Nullstellensatz. Despite being an observation about roots of a polynomial in n variables, it finds a usage in numerous fields - from Combinatorial Number Theory to Graph Theory. The theory itself is simple, but can be used in ingenious ways - appearing even as almost a necessary step to solve a problem during the 2007 IMO competition. During this time slot I will present a theorem and prove it with as I believe a simpler proof constructed by Mateusz Michałek, that is using a basic induction idea over a polynomial degree. Finally we will again follow the original N. Alon paper to see applications of a theorem in the graph coloring problems and some more.
(the seminar will only be online) |
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07.10.21628 Mikołaj Bojańczyk University of Warsaw |
Informatyka Teoretyczna Recognisable languages over monads |
Algebraic language theory originated in the study of regular languages via semigroups, instead of automata. The advantage of the semigroup approach is a richer structural theory, e.g. Green’s theory or the Factorisation Forest Theorem. (In contrast, the structural analysis of automata seldom goes beyond such elementary notions as “cycle” or “connected component”.) In this talk, I will discuss a more abstract view on semigroups, as Eilenberg-Moore algebras over the monad of finite words (aka the list monad in programming languages). Using this abstract view, by changing the monad, one can get the appropriate notion of “semigroup” for objects beyond finite words, e.g. trees or graphs. Sometimes, even theorems can be proved using this abstract view.
This talk is based on the draft monograph
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14.06.87338 Andrzej Dorobisz |
Informatyka Teoretyczna Local Computation Algorithms for Coloring of Uniform Hypergraphs |
We present a progress on local computation algorithms for two coloring of k-uniform hypergraphs. We focus on instances that (for a parameter α) satisfy strengthened assumption of Local Lemma of the form 21-αk(Δ+1)e<1, where Δ is the bound on the maximum edge degree of the hypergraph. We discuss how previous works on the subject can be used to obtain an algorithm that works in polylogarithmic time per query for α up to about 0.139. Then, we present a procedure that, within similar bounds on running time, solves wider range of instances by allowing α to be at most about 0.227. Joint work with Jakub Kozik |
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04.01.70911 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Local Dimension of Planar Poset |
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetril and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height. This is a joint work with Jarosław Grytczuk and W.T. Trotter. (the seminar will only be online) |
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06.02.68173 Marcin Pilipczuk University of Warsaw |
Informatyka Teoretyczna Recent progress in understanding H-free graphs for H being a path or a subdivided claw |
Graph classes excluding a path or a subdivided claw as an induced subgraph are so far mysterious: on one hand, beside some corner cases, they do not seem to have any good structural description, but on the other hand, a number of combinatorial problems - including Maximum Independent Set (MIS) - lack an NP-hardness proof in these graph classes, indicating a possible hidden structure that can be used algorithmically. Furthermore, graphs excluding a fixed path as an induced subgraph were one of the earliest examples of a chi-bounded graph class, with an elegant proof technique dubbed the Gyarfas' path argument. In the recent years the progress accelerated, leading to, among other results, (a) a quasi-polynomial-time algorithm for MIS in graphs excluding a fixed path as an induced subgraph, (b) a QPTAS for MIS in graphs excluding a subdivided claw as an induced subgraph, (c) the proof of the Erdos-Hajnal property in graph classes excluding a fixed forest and its complement. In the talk, I will survey these results, showing the role of the Gyarfas' path argument in most (all?) of them, and outline research directions for the future. |
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29.08.51745 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna The 1/3 - 2/3 conjecture |
A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field. (the seminar will only be online) |
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03.10.49007 Stefan Felsner Technische Universität Berlin |
Informatyka Teoretyczna Combinatorics of Pseudocircle Arrangements |
In this talk we survey results for pseudocircle arrangements. Along the way we present several open problems. Among others we plan to touch the following topics: * The taxonomy of classes of pseudocircle arrangements. The talk includes work of Grünbaum, Snoeyink, Pinchasi, Scheucher, myself, and others. |
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23.04.32580 Jędrzej Hodor |
Optymalizacja Kombinatoryczna Polynomial Treedepth Bounds in Linear Colorings |
Centered graph coloring is graph coloring, such that for every connected subgraph, this subgraph contains a vertex with a unique color (we call such a vertex center). Linear coloring is coloring, such that every path has a center. We denote by cen(G) and lin(G) respectively, a minimal number of colors needed. It is obvious that lin(G) ≤ cen(G). What about the other direction? Authors of the paper show that cen ≤ f(lin), where f is a polynomial of the degree 190. Moreover, the authors conjecture that cen ≤ 2 lin for every graph. What is interesting, we don't know how to prove such abound even for trees. Luckily, for some classes of graphs, we can do better than 190-poly. During the seminar, I will present results for simple classes of graphs and I will try to sketch the general proof. In particular, I will present a cubic bound for interval graphs. The proof in the paper is incorrect, but I and dr Micek managed to fix it. Finally, I will present our new result for graphs with bounded path width.
(the seminar will only be online) |
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28.05.29842 Bartosz Walczak |
Informatyka Teoretyczna Approximating Pathwidth for Graphs of Small Treewidth |
We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of O(t √ log t). This is the first algorithm to achieve an f(t)-approximation for some function f. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth Ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k. Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t' in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t'+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.
Joint work with Carla Groenland, Gwenaël Joret, and Wojciech Nadara. |
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07.12.70910 Kamil Kropiewnicki |
Optymalizacja Kombinatoryczna Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming |
We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems. This presentation might be of interest for, among the others, the participants of the Algorithmic Game Theory course as it presents the modern approach for maximizing revenue in second-price auctions.
(the seminar will only be online) |
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04.11.79147 Rafał Burczyński |
Optymalizacja Kombinatoryczna Bollobás-Eldridge-Catlin Conjecture on graph packing |
Let G1, G2 be n-vertex graphs. We say that they pack if they are edge-disjoint subgraphs of a complete graph on n vertices. The Bollobás-Eldridge-Catlin conjecture states that if (Δ(G1) + 1) (Δ(G2) + 1) < n + 1, then G1 and G2 pack. During the seminar, we will take a look at current results related to this problem, i.e. classes of graphs for which it has been proven as well as similar conjectures stemming from it. (the seminar will only be online) |
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27.07.79124 Weronika Lorenczyk |
Optymalizacja Kombinatoryczna The Cap Set Conjecture |
The cap set problem asks how large can a subset of Z/3Zn be and contain no lines or, more generally, how can large a subset of Z/pZn be and contain no arithmetic progression. The problem was open until 2016 when its basic version was solved. During the lecture, we'll see the motivation for thinking about this. It appears there are some interesting applications of this result in combinatorics, geometry, and even board games. (the seminar will only be online) |
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29.06.59982 Bartosz Wodziński |
Optymalizacja Kombinatoryczna Graph Removal Lemma |
Let H be a graph on h vertices. The Graph Removal Lemma states that for any ε > 0, there exists a constant δ(ε, H) > 0 such that for any n-vertex graph G with fewer than δnh subgraphs isomorphic to H, it is possible to eliminate all copies of H by removing at most εn2 edges from G. It has several important consequences in number theory, discrete geometry, graph theory, and computer science. During the seminar, I will discuss this lemma and its extensions. I will also tell about some of its applications, such as graph property testing and Szeremedi's Theorem proof.
(the seminar will only be online) |
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22.03.59959 Artur Kasymov |
Optymalizacja Kombinatoryczna Machine learning in Combinatorial Optimization |
Machine learning has already leaked almost all areas. What about Combinatorial Optimization? At this seminar, I will present basic ML concepts and methods in CO: Where you can add ML black box in your algorithm? Can heuristics be compared to ML? What are the recent achievements? (the seminar will only be online) |
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19.10.57111 Weronika Loreńczyk |
Podstawy Informatyki The Fractal Dimension of SAT Formulas by Carlos Ansotegui, Maria Bonet , Jesus Giraldez-Cru and Jordi Levy |
| Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise definition of the notion of structure. Recently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them. We study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances. We show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is affected by the addition of learnt clauses during the execution of SAT solvers. |
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21.02.40817 Bruno Pitrus |
Optymalizacja Kombinatoryczna Seven trees in one: objects of categories as complex numbers |
Consider the following absurd argument concerning planar, binary, rooted, unlabelled trees. Every such tree is either the trivial tree or consists of a pair of trees joined together at the root, so the set T of trees is isomorphic to 1+T². Pretend that T is a complex number and solve the quadratic T = 1+T² to find that T is a primitive sixth root of unity and so T⁶ = 1. Deduce that T⁶ is a one-element set; realize immediately that this is wrong. Notice that T⁷ = T is, however, not obviously wrong, and conclude that it is therefore right. In other words, conclude that there is a bijection T⁷ ≅ T built up out of copies of the original bijection T ≅ 1+T²: a tree is the same as seven trees.
(the seminar will only be online) |
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14.11.40793 Krzysztof Pióro |
Optymalizacja Kombinatoryczna Gallai’s conjecture |
A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌈n/2⌉. Gallai’s Conjecture has been verified for many classes of graphs. In this seminar, we will cover some of these graph classes. (the seminar will only be online) |
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03.05.37946 Maciej Nemś |
Podstawy Informatyki Regular Matching and Inclusion on Compressed Tree Patterns with Context Variables by Iovka Boneva, Joachim Niehren, and Momar Sakho |
| We study the complexity of regular matching and inclusion for compressed tree patterns extended by context variables. The addition of context variables to tree patterns permits us to properly capture compressed string patterns but also compressed patterns for unranked trees with tree and hedge variables. Regular inclusion for the latter is relevant to certain query answering on Xml streams with references. |
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09.07.21628 Szymon Żak |
Optymalizacja Kombinatoryczna Aleph: Efficient Atomic Broadcast in Asynchronous Networks with Byzantine Nodes |
In this seminar, I will cover general ideas that stand behind Aleph protocol. Aleph is a leaderless, fully asynchronous, Byzantine fault tolerant consensus protocol for ordering messages exchanged among processes. It is based on a distributed construction of a partially ordered set and the algorithm for reaching a consensus on its extension to a total order.
(the seminar will only be online) |
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12.10.49030 Jan Mełech |
Optymalizacja Kombinatoryczna Hamiltonian paths/cycles in vertex-transitive/symmetric graphs |
Graph is vertex-transitive if every vertex has the same local environment, so that no vertex can be distinguished from any other based on the vertices and edges surrounding it. In 1969, Lovasz conjectured that every finite connected vertex-transitive graph has Hamiltonian path. Moreover, up to now there are currently only five known connected vertex-transitive graphs not containing Hamiltonian cycle. In this seminar we will focus also on some other weaker variants of Lovasz conjecture related to other interesting class of graphs that encode the abstract structures of a groups - Cayley graphs. (the seminar will only be online) |
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05.07.49007 Mateusz Kaczmarek |
Optymalizacja Kombinatoryczna From linear lambda terms to rooted trivalent maps |
Recent work on the combinatorics of the linear lambda term shows that it has various connections to the theory of graph surfaces (maps). Based on paper [1] I will present a bijection between linear lambda terms (presented as diagrams) and rooted trivalent maps. Also, I will cover the recent conjecture proposed in 2019 that a special class of planar lambda terms can be counted the same way that rooted bicubic maps.
(the seminar will only be online) |
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31.01.46160 Weronika Loreńczyk - canceled |
Podstawy Informatyki The Fractal Dimension of SAT Formulas by Carlos Ansotegui, Maria Bonet , Jesus Giraldez-Cru and Jordi Levy |
| Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise definition of the notion of structure. Recently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them. We study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances. We show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is affected by the addition of learnt clauses during the execution of SAT solvers. |
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06.06.29865 Wojciech Buczek |
Optymalizacja Kombinatoryczna Inscribed square problem |
Let C be a Jordan curve. We say that polygon P is inscribed in C if all vertices of P belong to C. In the inscribed square problem we ask if every Jordan curve admits an inscribed square. It's also known as "Toeplitz’s conjecture" or the "Square peg problem". In this seminar, we will show some equivalent problems to this conjecture and focus on special cases of the Jordan curves. (the seminar will only be online) |
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27.02.29842 Bartłomiej Jachowicz |
Optymalizacja Kombinatoryczna Parameterized by treewidth algorithms for Hamiltonian Cycle |
The Hamiltonian Cycle problem is one of the oldest and most common NP-complete problems. It consists of checking whether in a given graph there is a cycle visiting each vertex exactly once. I will present a parameterized algorithm based on graph tree decomposition. Assuming that a nice tree decomposition of the width k is known at the input Hamiltonian cycle problem can be solved in a time 2(O(k))n(O(1)). (the seminar will only be online) |
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15.08.26994 Katarzyna Król |
Podstawy Informatyki A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods by Mirai Ikebuchi |
| It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier’s homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds. |
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30.01.10700 Michał Zwonek |
Optymalizacja Kombinatoryczna Approximate Distance Oracles |
Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v∈V, returns an approximation to the actual distance between u and v which is within some bounded stretch factor of the true distance. The first work in this area was done by Mikkel Thorup and Uri Zwick, they devised an oracle with construction time being O(kmn(1/k)) and with the space complexity of O(kn(1+1/k)). The achieved stretch, that is the measure of how accurate the answer by the approximate oracle will be, is bounded by (2k-1). The query time is O(k), this has been subsequently improved to O(log n) by Wulff-Nilsen and to O(1) by Shiri Chechik. (the seminar will only be online) |
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22.10.10676 Wojciech Grabis |
Optymalizacja Kombinatoryczna Double-critical graph conjecture |
A connected graph G is called double-critical if the chromatic number of G decreases by two if any two adjacent vertices of G are removed. In 1966, Erdős and Lovász conjectured that the only double-critical n-chromatic graph is the complete graph on n vertices. During the seminar, I will present what has been verified about the conjecture. (the seminar will only be online) |
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10.04.7829 Wojciech Węgrzynek |
Podstawy Informatyki The repetition threshold for binary rich words by James Currie, Lucas Mol and Narad Rampersad |
| A word of length n is rich if it contains n nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2 + \Sqrt{2}/2$ ( = 2.707) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2 + \Sqrt{2}/2$ ). In this article, we give a structure theorem for infinite binary rich words that avoid 14/5-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2 + \Sqrt{2}/2$ , as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.
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10.11.73671 Krzysztof Potępa |
Optymalizacja Kombinatoryczna Erdős–Hajnal conjecture |
A well-known theorem of Erdős states that there exists a graph on n vertices, with no clique or independent set of a size larger than O(log n). The Erdős–Hajnal conjecture says it is very different if we consider families of graphs defined by forbidden induced subgraphs. Specifically, it is conjectured that for every graph H, there exists a constant δ(H) such that every H-free graph G has either a clique or independent set of size |V(G)|δ(H). We will discuss some classes of graphs for which the conjecture has been proven, as well as weaker theorems that hold for all graphs. (the seminar will only be online) |
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02.08.73648 Marcin Serwin |
Optymalizacja Kombinatoryczna (m,n)-cycle cover conjectures |
An (m,n)-cycle cover is a covering of a graph consisting of m cycles such that every edge is covered exactly n times. For positive integers m, n it is natural to ask what family of graphs have (m,n)-cycle covers. The answers are known for some values, but for many others, they are conjectured or totally open. (the seminar will only be online) |
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19.01.70801 Wojtek Grabis |
Podstawy Informatyki (Optimal) Duplication is not Elementary Recursive by Andrea Asperti, Paolo Coppola and Simone Martini |
| In 1998 Asperti and Mairson proved that the cost of reducing a lambda-term using an optimal lambda-reducer (a la L´evy) cannot be bound by any elementary function in the number of shared-beta steps. We prove in this paper that an analogous result holds for Lamping’s abstract algorithm. That is, there is no elementary function in the number of shared beta steps bounding the number of duplication steps of the optimal reducer. This theorem vindicates the oracle of Lamping’s algorithm as the culprit for the negative result of Asperti and Mairson. The result is obtained using as a technical tool Elementary Affine Logic. |
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14.09.51635 Michał Zwonek |
Podstawy Informatyki A Confluent Rewriting System Having No Computable, One-Step, Normalizing Strategy by JAKOB GRUE SIMONSEN |
| A full and finitely generated Church-Rosser term rewriting system is presented that has no computable onestep, normalizing strategy; the system is both left- and right-linear. The result provides a negative answer to a question posed by Kennaway in 1989: Number 10 on the List of Open Problems in Rewriting. |
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21.11.35317 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Harmonious Coloring of Hypergraphs |
A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge share the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. We prove that k-uniform hypergraphs of bounded maximum degree Δ satisfy h(H) = O(k√k!m), where m is the number of edges in H which is best possible up to a multiplicative constant. Moreover, for every fixed Δ, this constant tends to 1 with k → ∞. We use a novel method, called entropy compression, that emerged from the algorithmic version of the Lovász Local Lemma due to Moser and Tardos. This is joint work with Sebastian Czerwinski, Jarosław Grytczuk, and Paweł Rzazewski. (the seminar will only be online) |
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17.02.35263 Dzianis Pivavarau, Dominik Wielgórski |
Explicit two-deletion codes with redundancy matching the existential bound |
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16.07.16152 Piotr Mikołajczyk |
Optymalizacja Kombinatoryczna Polynomial algorithms for CFGs via semiring embeddings |
A few years ago M. Might et al. published somehow unusual approach to parsing context-free grammars by using derivative operator. Later it was proven, that its worst case complexity is polynomial, putting it on a par with other classical approaches. We introduce an elegant generalization to this method by a generic algorithm parametrized with a semiring. Depending on the chosen algebra we can obtain polynomial algorithms for problems like parsing, recognizing or counting parse trees for CFGs. (the seminar will only be online) |
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12.10.16097 Bartłomiej Jachowicz, Mateusz Kaczmarek |
Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs |
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02.01.13305 Przemysław Simajchel |
Podstawy Informatyki COMPLEXITY PROBLEMS IN ENUMERATIVE COMBINATORICS by IGOR PAK |
| The paper gives a broad survey of recent results in Enumerative Combinatorics and their complexity aspects. |
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09.09.79014 CANCELED |
Podstawy Informatyki COMPLEXITY PROBLEMS IN ENUMERATIVE COMBINATORICS by IGOR PAK |
| The paper gives a broad survey of recent results in Enumerative Combinatorics and their complexity aspects. |
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15.11.62696 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Conjecture 1/3 - 2/3 |
A given pair of two incomparable elements x, y in poset P is called balanced if, of all line extensions P, the element x lies above y by at most 2/3 and on at least 1/3 of all extensions of the poset P. The 1/3 - 2/3 conjecture says that any poset that is not linear has a balanced pair. The talk presents basic definitions and an overview of the most important results in this field. (the seminar will only be online) |
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12.02.62642 Marcin Serwin, Wojciech Buczek |
A Double-Exponential Lower Bound for the Distinct Vectors Problem |
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14.06.59849 Piotr Mikołajczak |
Podstawy Informatyki Asymptotic Approximation by Regular Languages by Ryoma Sin’ya |
| This paper investigates a new property of formal languages called REG-measurability where REG is the class of regular languages. Intuitively, a language L is REG-measurable if there exists an infinite sequence of regular languages that “converges” to L. A language without REG-measurability has a complex shape in some sense so that it can not be (asymptotically) approximated by regular languages. We show that several context-free languages are REG-measurable (including languages with transcendental generating function and transcendental density, in particular), while a certain simple deterministic context-free language and the set of primitive words are REG-immeasurable in a strong sense. |
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12.07.43531 Vladyslav Rachek |
Optymalizacja Kombinatoryczna Small weak epsilon-nets |
Let P be a set of n points in R2, ε > 0. A set of points Q is called a weak ε-net for P with respect to a family S of objects (e.g. axis-parallel rectangles or convex sets) if every set from S containing more than εn points of P contains a point from Q. Let R be the family of all axis-parallel rectangles in R2 and εRk be the smallest real number such that for any P there exists a weak εRk-net of size k. The work by Aronov et al. suggests that the inequality εRk ≤ 2/(k+3) may hold. In this talk we present the complete proofs of this inequality for k=1,...,5 and prove that this bound is tight for k=1,2,3. Besides, it is not clear how to construct optimal nets. Langerman conjectured that k-point 2/(k+3)-nets can be chosen from some speciffc set of points which are the intersections of grid lines, where the grid is of size k×k. We give counterexamples to this conjecture for nets of size 3 through 6.
(the seminar will only be online) |
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07.10.43476 Krzysztof Pióro, Krzysztof Potępa |
Modular Subset Sum |
W problemie Modular Subset Sum dane są liczba naturalna m, n-elementowy multizbiór S liczb całkowitych z zakresu od 0 do m-1 oraz liczba t, dla której chcemy rozstrzygnąć, czy istnieje podzbiór S, który się do niej sumuje modulo m.
Przedstawimy własne algorytmy rozwiązujące powyższy problem. Wszystkie z nich będą sprowadzały problem Modular Subset Sum do problemu tekstowego. Na początku przedstawimy prosty algorytm działający w czasie O(n + m*log2(m)) wykorzystujący haszowanie i drzewa przedziałowe. Następnie pokażemy jak poprawić jego złożoność do O(n + m*log(m)). Na końcu zaprezentujemy w pełni deterministyczny wariant algorytmu działający w czasie O(n + m*log(m)*α(m)).
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07.02.40684 Jędrzej Hodor |
Podstawy Informatyki Bijective link between Chapoton’s new intervals and bipartite planar maps by Wenjie Fang |
| In 2006, Chapoton defined a class of Tamari intervals called “new intervals” in his enumeration of Tamari intervals, and he found that these new intervals are equienumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called “degree tree”. Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy. |
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06.03.24366 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna From the 1-2-3 Conjecture to the Riemann Hypothesis |
A series of open (and solved) problems will be presented at the seminar, starting with the 1-2-3 Conjecture and ending with the Riemann Hypothesis. (the seminar will only be online) |
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21.07.24252 Patryk Mikos |
Informatyka Teoretyczna Geometric and weight constraints in Online Interval Coloring |
| PhD defense - room 0004 |
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05.12.29864 Bartosz Wodziński |
Optymalizacja Kombinatoryczna On the unique games conjecture |
For many hard problems, instead of solving them directly, we need good approximation algorithms. Apart from good their time complexity and decent approximation factor guarantee, we would like to know whether they achieve the best possible approximation ratio (assuming P ≠ NP) possible. Unfortunately, for many NP-complete problems, there is a huge gap between best-known approximation ratio and the ratio that is proved to be unachievable in polynomial time. For instance, for Vertex Cover problem, we don't know any algorithm having a better ratio than 2, and it has been proved in 2005 that it is impossible to get a better ratio than ~1.36. As an attempt to fill in this gap, in 2002, the so-called Unique Games Conjecture was formulated by Khot. It states that having a (1-𝜀)-satisfiable instance of Unique Label Cover problem, it is NP-hard to find a solution satisfying even epsilon fraction of constraints. Assuming it, we are able to prove many tight inapproximability results, for example, it implies that Goemans-Williamson Algorithm for Max-Cut problem achieves the best possible approximation rate. It also follows that we cannot get any better ratio than 2 in the case of Vertex Cover problem. The Unique Games Conjecture is an unusual open problem since the academic world is about evenly divided on whether it is true or not. During the seminar, I will cover this conjecture in more details giving more examples of its influence and presenting recent progress in order to prove it.
(the seminar will only be online) |
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28.08.29841 Gabriela Czarska |
Optymalizacja Kombinatoryczna The Lonely Runner Conjecture |
Abstract. Suppose that k runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least 1/k from all the other runners. We prove that with probability tending to one, a much stronger statement holds for random sets in which the bound 1/k is replaced by 1/2 − ε. The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs. (the seminar will only be online) |
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21.02.29732 Wojciech Grabis |
Podstawy Informatyki Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup |
| This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|_gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus. |
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21.02.29732 Wojciech Grabis |
Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup |
| This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus. |
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21.02.29732 Wojciech Grabis |
Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup |
| This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus. |
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21.02.29732 Wojciech Grabis |
Decidability of regular language genus computation by Guillaume Bonfante and Florian L. Deloup |
| This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|_gen to be the minimal size of all finite deterministic automata of genus g(L) computing L. We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|_gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus. |
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27.06.13437 Paweł Mader |
Optymalizacja Kombinatoryczna Oblivious routing on 2d grid |
Oblivious routing is a routing problem, in which a packet path is selected independently from path choices of other packets. One of the open problems is to find networks for which there exists an oblivious routing algorithm, which allows simultaneously optimizing stretch and congestion of the network. We are presenting an algorithm for oblivious routing on 2dgrid, which is O(log n) approximation for congestion and Θ(1) approximation of stretch. (the seminar will only be online) |
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20.03.13414 Raja L'hamri Mohammed V University |
Optymalizacja Kombinatoryczna Examples of codes from zero-divisor graphs |
In 2013, Dankelmann, Key, and Rodrigues introduced and investigated codes from incidence matrices of a graph. Several authors have been developed their study to several context. In this talk, we present some properties of codes associated with zero divisor graphs. Recall, the zero divisor graph of R denoted by Γ(R), is the simple graph associated with R whose set of vertices consists of all nonzero zero-divisors of R such that two distinct vertices x and y are joined by an edge if xy = 0. This is joint work with K. Abdelmoumen, D. Bennis, and F. Taraza.
(the seminar will only be online) |
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16.10.10566 Ruslan Yevdokymov |
Podstawy Informatyki Learnability can be undecidable by Shai Ben-David, Pavel Hrubes, Shay Moran, Amir Shpilka and Amir Yehudayoff |
| The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression. |
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04.03.79147 Michał Stobierski |
Optymalizacja Kombinatoryczna The 1-2-3 Conjecture |
We all know how important mathematical theorems are in general. Less obvious is the fact that theorems in one area like algebra or number theory could have a significant impact on another. In our case, these will be combinatorial problems. In this presentation, We will go through a few simple graph coloring questions (based on the original 1-2-3 Conjecture), which unfortunately don't have simple solutions at all and we'll classify them. Moreover, thanks to Combinatorial Nullstellensatz and some greedy techniques, we will be able to prove some weaker versions of our original claims. And finally, we will see how one simple question, through a chain of small modifications, can lead us to completely different problems. (the seminar will only be online) |
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25.11.79123 Rafał Byczek |
Optymalizacja Kombinatoryczna Wegner’s conjecture - colouring the square of a planar graph |
The square G2 of a graph G is the graph with the same vertex set in which two vertices are joined by an edge if their distance in G is at most two. The chromatic number of the square of a graph G is between D + 1 and D2 + 1, where D is the maximum degree of G. Equivalently, the square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In 1977, Gerd Wegner proved that the square of cubic planar graphs is 8-colorable. He conjectured that his bound can be improved - the chromatic number of G2 is at most 7, if D = 3, at most D + 5, if 4 ≤ D ≤ 7, and [3D / 2] + 1, otherwise. Wegner also gave some examples to illustrate that these upper bounds can be obtained. C. Thomassen (2006) proved the conjecture is true for planar graphs with D = 3. The conjecture is still open for planar graphs with D ≥ 4. However several upper bounds in terms of maximum degree D have been proved as follows. In 1993, Jonas proved that χ(G2) ≤ 9D-19, for planar graphs with D ≥ 5. Agnarsson and Halldorson showed that for every planar graph G with maximum degree D ≥ 749, χ(G2) ≤ [9D / 5] + 2. Van den Heuvel and McGuinness (2003) showed that χ(G2) ≤ 2D + 25, Bordin (2002) proved that χ(G2) ≤ [9D / 5] + 1, if D ≥ 47, and Molloy and Salavatipour (2005) proved χ(G2) ≤ [5D / 3] + 78, moreover, χ(G2) ≤ [5D / 3] + 25 if D ≥ 241. Moreover, conjecture is confirmed in the case of outerplanar graphs and graphs without K4 minor. The aim of the seminar will be to present the main facts about Wegner’s conjecture and main techniques and ideas which were used to prove some upper bounds. The presentation will be based on my master thesis. (the seminar will only be online) |
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14.05.76272 Szymaon Kapała |
Podstawy Informatyki Searching for shortest and least programs by Cristian Caludea, Sanjay Jain, Wolfgang Merkle and Frank Stephan |
| The Kolmogorov complexity of a string x is defined as the length of a shortest program p of x for some appropriate universal machine U, that is, U(p) =x and p is a shortest string with this property. Neither the plain nor the prefix-free version of Kolmogorov complexity are recursive but for both versions it is well-known that there are recursive exact Solovay functions, that is, recursive upper bounds for Kolmogorov complexity that are infinitely often tight. Let a coding function for a machine M be a function f such that f(x) is always a program of x for M. From the existence of exact Solovay functions it follows easily that for every universal machine there is a recursive coding function that maps infinitely many strings to a shortest program. Extending a recent line of research, in what follows it is investigated in which situations there is a coding function for some universal machine that maps infinitely many strings to the length-lexicographically least program. The main results which hold in the plain as well as in the prefix-free setting are the following. For every universal machine there is a recursive coding function that maps infinitely many strings to their least programs. There is a partial recursive coding function (defined in the natural way) for some universal machine that for every set maps infinitely many prefixes of the set to their least programs. Exactly for every set that is Bennett shallow (not deep), there is a recursive coding function for some universal machine that maps all prefixes of the set to their least programs. Differences between the plain and the prefix-free frameworks are obtained by considering effective sequences I_1, I_2, ...of mutually disjoint finite sets and asking for a recursive coding function for some universal machine that maps at least one string in each set I_n to its least code. Such coding functions do not exist in the prefix-free setting but exist in the plain setting in case the sets I_n are not too small. |
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20.07.59958 Wojtek Grabis |
Optymalizacja Kombinatoryczna Algorithms for Destructive Shift Bribery. |
Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods.
(the seminar will only be online) |
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16.02.57111 Piotr Mikołajczyk |
Podstawy Informatyki Lambda Calculus and Probabilistic Computation by Claudia Faggian and Simona Ronchi della Rocca |
| We introduce two extensions of the lambda -calculus with a probabilistic choice operators, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic ! which allows us to develop a unified, modular approach. |
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21.06.40816 Jan Mełech |
Optymalizacja Kombinatoryczna Upper Bounds for the domination numbers of graphs |
Sharareh Alipour and Amir Jafari showed various upper bounds for minimal cardinality of (a,b)-dominating set. For positive integers a and b, a subset S ⊆ V(G) is an (a,b)-dominating set if every vertex v ∈ S is adjacent to at least a vertices inside S and every vertex v ∈ V\S is adjacent to at least b vertices inside S. To achieve upper bounds, the authors used Turan's Theorem and Lovasz Local Lemma. These tools allowed them to obtain well-known bounds in a simpler way or new improved bounds in some special cases, including regular graphs.
(the seminar will only be online) |
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14.03.40793 Szymon Kapała |
Optymalizacja Kombinatoryczna Goldbach conjectures (weak and strong). |
(the seminar will only be online) |
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11.10.37945 Przemysław Simajchel |
Podstawy Informatyki Dance of the Starlings by Henk Barendregt, Jorg Endrullis, Jan Klop and Johannes Waldmann |
| In this birdwatching paper our binoculars are focused upon a particular bird from Smullyan's enchanted forest of combinatory birds, to wit the Starling. In the feathers of lambda -calculus this bird has the plumage \abc:ac(bc). This term is usually named S, reminiscent of its inventor Schonfinkel and also the combinatory ornithologist Smullyan. The combinator S is important for a variety of reasons. First, it is part of the \{ S, K\} basis for Combinatory Logic (CL). Second, there are several interesting questions and observations around S, mostly referring to termination and word problems. Our paper collects known facts, but poses in addition several new questions. For some of these we provide solutions, but several tough open questions remain. |
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07.11.21627 Michał Zwonek |
Optymalizacja Kombinatoryczna 3-flow conjecture |
3-flow-conjecture Grötzsch proved that every triangle free (and loopless) planar graph is 3-colorable. By flow/coloring duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. The 3-flow conjecture asserts that this is still true without the assumption of planarity. Jaeger proved that 4-edge-connected graphs have nowhere-zero 4-flows. The following weak version of the 3-flow conjecture used to remain open until 2010, but the original 3-flow conjecture remains wide open. C̶o̶n̶j̶e̶c̶t̶u̶r̶e̶ (The weak 3-flow conjecture (Jaeger)) These problems and the surrounding results will be presented during the seminar.
(the seminar will only be online) |
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05.06.18780 Bartłomiej Puget |
Podstawy Informatyki Evidence Normalization in System FC by Dimitrios Vytiniotis and Simon Peyton Jones |
| System FC is an explicitly typed language that serves as the target language for Haskell source programs. System FC is based on System F with the addition of erasable but explicit type equality proof witnesses. Equality proof witnesses are generated from type inference performed on source Haskell programs. Such witnesses may be very large objects, which causes performance degradation in later stages of compilation, and makes it hard to debug the results of type inference and subsequent program transformations. In this paper, we present an equality proof simplification algorithm, implemented in GHC, which greatly reduces the size of the target System FC programs. |
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26.11.84622 Mateusz Kaczmarek |
Optymalizacja Kombinatoryczna χ-boundedness |
If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? To answer that question Paul Seymour and Alex Scott took a closer look at recent progress in this field in their χ-boundedness survey. Based on their work I will present some results on forests and holes and few open problems like Gyárfás-Sumner conjecture or χ-boundedness connection to Erdős-Hajnal conjecture.
(the seminar will only be online) |
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18.08.84599 Kornel Dulęba |
Optymalizacja Kombinatoryczna Odd Perfect numbers |
A number is perfect if it is equal to the sum of its divisors. So far only even perfect numbers have been found. For example, it was proven that squares of Mersenne’s numbers are perfect. However, no one has been able to prove that odd perfect numbers don’t exist. I’m going to start by presenting a summary of known facts about odd prime numbers. Then I’ll prove that an odd perfect number with at least eight distinct prime factors has to be divisible by 5.
(the seminar will only be online) |
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16.03.81752 Jakub Dyczek |
Podstawy Informatyki On probabilistic term rewriting by Martin Avanzinia,Ugo Dal Lago and Akihisa Yamadac |
| We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method polynomial and matrix interpretations are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by means of multidistribution reduction sequences, thus accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule. |
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21.07.65457 Bartłomiej Jachowicz |
Optymalizacja Kombinatoryczna Lonely runner conjecture |
One of number theory open problem is the Lonely Runner Conjecture. It is interesting for several reasons. First the conjecture is relatively intuitive to grasp and easy to state. This conjecture can be find in two different contexts: as a problem in Diophantine’s approximation and as a geometric view obstruction problem. What is more, the difficulty of proving the Lonely Runner Conjecture may seem to increase exponentially with the number of runners. I present statement of the conjecture and known partial results.
(the seminar will only be online) |
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13.04.65434 Filip Bartodziej |
Optymalizacja Kombinatoryczna Meyniel’s conjecture on the cop number |
A cops and robbers problem determines if the number of cops is sufficient to always catch a robber in a game with defined rules played on an undirected graph. Cop number of a graph is the minimal number of cops necessary for cops to win in that game on the specific graph. Mayniel’s conjuncture remains an open problem and states that cop number for graphs of order n is sqrt(n). I’ll present a survey of results achieved that are related to this conjecture.
(the seminar will only be online) |
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09.11.62586 Jan Kościsz |
Podstawy Informatyki Fast Synchronization of Random Automata by Cyril Nicaud |
| A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerný conjectured in 1964 that if a n-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n − 1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^3 n) with high probability. As a consequence, this proves that most automata satisfy the Cerný conjecture. |
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15.03.46292 Mateusz Pabian |
Optymalizacja Kombinatoryczna Synchronizing Automata and the Černý Conjecture |
I present many results and finally open problem related to synchronizing automata and synchronizing word sends any state of the DFA to one and the same state. This leads to the some natural problems such as: how can we restore control over such a device if we don't know its current state but can observe outputs produced by the device under various actions? I prove some uperbounds for length of this kind of word and in particular I will make a statement of Cerny conjecture.
(the seminar will only be online) |
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06.12.46268 Adrian Siwiec |
Optymalizacja Kombinatoryczna Online Computation with Untrusted Advice |
The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. We study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner.To this end, we focus on well-studied online problems such as ski rental, online bidding, bin packing, and list update.
(the seminar will only be online) |
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05.07.43421 Magdalena Proszewska |
Podstawy Informatyki Singular value automata and approximate minimization by Borja Balle, Prakash Panangaden and Doina Precup |
| The present paper uses spectral theory of linear operators to construct approximately minimal realizations of weighted languages. Our new contributions are: (i) a new algorithm for the singular value decomposition (SVD) decomposition of finite-rank infinite Hankel matrices based on their representation in terms of weighted automata, (ii) a new canonical form for weighted automata arising from the SVD of its corresponding Hankelmatrix, and (iii) an algorithm to construct approximate minimizations of given weighted automata by truncating the canonical form. We give bounds on the quality of our approximation. |
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09.11.27126 Wojciech Buczek |
Optymalizacja Kombinatoryczna Seymour's Second Neighbourhood Conjecture |
Seymour's Second Neighbourhood Conjecture tells us, that any oriented graph has a vertex whose outdegree is at most its second outdegree, which is also known as Second neighborhood problem. Intuitively, it suggests that in a social network described by such a graph, someone will have at least as many friends-of-friends as friends. We will say about Chen-Shen-Yuster prove, that for any digraph D, there exists a vertex v such that |N++(v)|≥γ|N+(v)|, where γ=0.67815. We will consider graphs, in which we know, that such vertex exists. We will also say about unsuccessful attempts at proving this conjecture.
(the seminar will only be online) |
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02.08.27103 Mikołaj Twaróg |
Optymalizacja Kombinatoryczna Collatz conjecture |
The Collatz conjecture, also known as 3n + 1 conjecture considers a function, which returns n/2 if n is even and 3n + 1 if n is odd. The conjecture states that for every n we can repeatedly apply this function to eventually reach 1. I will talk about different approaches to proving this conjecture. (the seminar will only be online) |
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28.02.24256 Jacek Kurek |
Podstawy Informatyki Complexity of translations from resolution to sequent calculus by GISELLE REIS and BRUNO PALEO |
| Resolution and sequent calculus are two well-known formal proof systems. Their differences make them suitable for distinct tasks. Resolution and its variants are very efficient for automated reasoning and are in fact the theoretical basis of many theorem provers. However, being intentionally machine oriented, the resolution calculus is not as natural for human beings and the input problem needs to be pre-processed to clause normal form. Sequent calculus, on the other hand, is a modular formalism that is useful for analysing meta-properties of various logics and is, therefore, popular among proof theorists. The input problem does not need to be pre-processed, and proofs are more detailed. However, proofs also tend to be larger and more verbose. When the worlds of proof theory and automated theorem proving meet, translations between resolution and sequent calculus are often necessary. In this paper, we compare three translation methods and analyse their complexity. |
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04.07.7961 Adam Pardyl |
Optymalizacja Kombinatoryczna Undirected edge geography |
The game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex, erasing the edge in between. The player who first has no legal move loses the game. We analyze the space complexity of the decision problem of determining whether a start position in this game is a win for the first player. We also show a polynomial time algorithm for finding winning moves for bipartite graphs.
(the seminar will only be online) |
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27.03.7938 Piotr Mikołajczyk |
Optymalizacja Kombinatoryczna ARRIVAL game |
Consider a directed graph such that every vertex has at most 2 outgoing edges - one of them is labeled as 'open' (we can traverse it) and the second one is labeled as 'closed' (we cannot traverse it). Every time we go somewhere from the vertex v, labels at its two edges are swapped, so the next time we visit v, we will take another direction. Given designated two vertices: origin and destination, we need to decide, whether eventually we will reach destination starting in the origin. This problem lies in both NP and coNP, but it is still an open question whether it belongs to P.
(the seminar will only be online) |
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23.10.5090 Rafał Byczek |
Podstawy Informatyki Bijection between oriented maps and weighted non-oriented maps by Agnieszka Czyzewska-Jankowska and Piotr Śniady |
| We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitrary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. |
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01.12.73647 Vladyslav Rachek |
Optymalizacja Kombinatoryczna Small weak epsilon-nets |
Let P be a set of n points in R2. A point q (not necessarily in P) is called a centerpoint of P if each closed half-plane containing q at least ⌈n/3⌉ points of P, or, equivalently, any convex set that contains more than 2/3 n points of P must also contain q. It is a well-known fact that a centerpoint always exists and the constant 2/3 is the best possible. Can we improve this constant by using, say, two points, or some other small number of points? In the presentation we'll try to answer those questions. Vladyslav Rachek. Small weak epsilon-nets. slides. 2020. (the seminar will only be online) |
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29.06.70800 Michał Zwonek |
Podstawy Informatyki FUNCTIONAL PEARL How to find a fake coin by RICHARD BIRD |
The aim of this pearl is to solve the following well-known problem that appears in many puzzle books, for example Levitin & Levitin (2011) and Bellos (2016), usually for the particular case n=12.
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03.11.54505 Kamil Rajtar |
Optymalizacja Kombinatoryczna How voting can be manipulated during selecting voting places |
During today presentation we will learn how we can use graph theory to proof hardness of general problem of manipulating poll outcome. Based on paper: "Selecting Voting Locations for Fun and Profit" written by Zack Fitzsimmons and Omer Lev. Zack Fitzsimmons, Omer Lev. Selecting Voting Locations for Fun and Profit. arXiv:2003.06879. 2020. (the seminar will only be online) |
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26.07.54482 Mateusz Tokarz |
Optymalizacja Kombinatoryczna The Hadwiger-Nelson problem |
We will focus on Hadwiger-Nelson problem - an open question from geometric graph theory that asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. There are a few interesting theorems related to the problem and results which we will go through. We will focus in particular on the most recent result of Aubrey de Grey who showed that the desired chromatic number is at least 5.
(the seminar will only be online) |
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22.02.51635 Mateusz Tokarz, wyniki własne, kontynuacja |
Podstawy Informatyki The largest fixed point in iterative programs |
| We study the smallest ordinal number α such that every Prologue program will reach its greatest fixed point after α downward iterations. Firstly, we show that the continuity of Prologue’s resolution function does not help with this matter. Then, due to the embedding of the recursive functions in Prologue, we get that α is at least Church-Kleene Omega. Using recursive linear order presented in “On the Forms of the Predicates in the Theory of Constructive Ordinals“ (Kleene, 1944) we construct a Prologue’s program requiring at least C-K Omega steps to achieve its greatest fixed point. To get the upper bound on α we use clockable ordinals introduced in “Infinite Time Turing Machines” (Joel David Hamkins, Andy Lewis, 1998). |
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17.10.32469 Mateusz Tokarz wyniki własne |
Podstawy Informatyki The largest fixed point in iterative programs |
| We study the smallest ordinal number α such that every Prologue program will reach its greatest fixed point after α downward iterations. Firstly, we show that the continuity of Prologue’s resolution function does not help with this matter. Then, due to the embedding of the recursive functions in Prologue, we get that α is at least Church-Kleene Omega. Using recursive linear order presented in “On the Forms of the Predicates in the Theory of Constructive Ordinals“ (Kleene, 1944) we construct a Prologue’s program requiring at least C-K Omega steps to achieve its greatest fixed point. To get the upper bound on α we use clockable ordinals introduced in “Infinite Time Turing Machines” (Joel David Hamkins, Andy Lewis, 1998). |
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12.01.65434 Jan Gwinner |
Optymalizacja Kombinatoryczna Spectrally Robust Graph Isomorphism |
In the paper authors consider certain variants of Graph Isomorphism problem. They focus on properties of graph spectra and eigenspaces - namely if Laplacian of one of the graphs is greater or equal to another in Loewner ordering. In the first part of the paper they prove that one of the problems named Spectral Graph Dominance is NPC. The rest of the paper is devoted to an approximation algorithm for special case of the problem called Spectrally Robust Graph Isomorphism. |
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10.08.62586 Weronika Grzybowska i Mateusz Tokarz |
Podstawy Informatyki On two subclasses of Motzkin paths and their relation to ternary trees by Helmut Prodinger, Sarah J. Selkirk and Stephan Wagner |
| Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed. |
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15.12.46291 Gabriela Czarska |
Optymalizacja Kombinatoryczna Driver surge pricing |
Authors study Uber's pricing mechanisms from the perspective of drivers, presenting the theoretical foundation that has informed the design of Uber’s new additive driver surge mechanism. They present a dynamic stochastic model to capture the impact of surge pricing on driver earnings and their strategies to maximize such earnings. Nikhil Garg, Hamid Nazerzadeh. Driver Surge Pricing. arXiv. 2019. |
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06.09.46268 Bartosz Podkanowicz |
Optymalizacja Kombinatoryczna Planar graphs have bounded queue-number |
The paper presents proof that the queue number of planar graphs is bounded. It also mentions generalizations of the result and other theorems that have similar proofs. |
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04.12.46213 Katarzyna Król, Paweł Mader |
On the Complexity of Lattice Puzzles [Kobayashi et al.] |
| Autorzy pracy badają złożoność obliczeniową tradycyjnej łamigłówki zwaną dalej układanką kratową. Celem układanki jest złożenie kraty o wymiarach n×n z 2n płytek z szczelinami. Łamigówka ta jest powszechnie znanym problemem, niemniej jednak do tej pory nie była ona badana przez informatykę teoretyczną. Autorzy pracy pokazują, że naturalne warianty tej układanki redukują się do podklas w klasie złożoności NP. Jedną z takich podklas jest klasa problemu izomorfizmów grafów GI. O ile wiadomo autorom pracy, jest to pierwszy nietrywialny GI-zupełny problem scharakteryzowany przez klasyczną łamigłówkę. |
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11.10.43530 Michał Seweryn |
Informatyka Teoretyczna Erdös-Hajnal properties for powers of sparse graphs |
The notion of nowhere dense classes of graphs attracted much attention in recent years and found many applications in structural graph theory and algorithmics. The powers of nowhere dense graphs do not need to be sparse, for instance the second power of star graphs are complete graphs. However, it is believed that powers of sparse graphs inherit somewhat simple structure. In this spirit, we show that for a fixed nowhere dense class of graphs, a positive real ε and a positive integer d, in any n-vertex graph G in the class, there are disjoint vertex subsets A and B with |A|=Ω(n) and |B|=Ω(n1-ε) such that in the d-th power of G, either there is no edge between A and B, or there are all possible edges between A and B.
Joint work with Marcin Briański, Piotr Micek and Michał Pilipczuk |
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05.04.43421 Wojciech Grabis |
Podstawy Informatyki Ant colony optimization theory: A survey by Marco Dorigoa and Christian Blumb |
| Research on a new metaheuristic for optimization is often initially focused on proof-of-concept applications. It is only after experimental work has shown the practical interest of the method that researchers try to deepen their understanding of the method’s functioning not only through more and more sophisticated experiments but also by means of an effort to build a theory. Tackling questions such as “how and why the method works’’ is important, because finding an answer may help in improving its applicability. Ant colony optimization, which was introduced in the early 1990s as a novel technique for solving hard combinatorial optimization problems, finds itself currently at this point of its life cycle. With this article we provide a survey on theoretical results on ant colony optimization. First, we reviewsome convergence results. Then we discuss relations between ant colony optimization algorithms and other approximate methods for optimization. Finally, we focus on some research efforts directed at gaining a deeper understanding of the behavior of ant colony optimization algorithms. Throughout the paper we identify some open questions with a certain interest of being solved in the near future. |
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10.08.27126 Wojtek Grabis |
Optymalizacja Kombinatoryczna Algorithms for Destructive Shift Bribery. |
Destructive Shift Bribery is a problem in which we are given an election with a set of candidates and a set of voters, a budget , a despised candidate and price for shifting the despised candidate in the voters rankings. Our objective is to ensure that selected candidate cannot win the election. We're going to study the complexity of this problem under diffrent election methods. Andrzej Kaczmarczyk, Piotr Faliszewski. Algorithms for Destructive Shift Bribery. arXiv. 2018. |
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03.05.27103 Dominik Gryboś |
Optymalizacja Kombinatoryczna Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice |
The paper shows that the Diffie-Hellman protocol is not as secure as we thought. The authors present the Logjam attack, which consists in quickly calculating discrete logarithms based on previously performed calculations. This can be done because many websites use the same prime numbers in the message encryption process. |
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29.11.24255 Piotr Gaiński |
Podstawy Informatyki How Similar Are Two Elections by Piotr Faliszewski, Piotr Skowron, Arkadii Slinko, Stanisław Szufa and Nimrod Talmon |
| We introduce the ELECTION ISOMORPHISM problem and a family of its approximate variants, which we refer to as d-ISOMORPHISM DISTANCE (d-ID) problems (where d is a metric between preference orders). We show that ELECTION ISOMORPHISM is polynomial-time solvable, and that the d-ISOMORPHISM DISTANCE problems generalize various classic rank-aggregation methods (e.g., those of Kemeny and Litvak). We establish the complexity of our problems (including their inapproximability) and provide initial experiments regarding the ability to solve them in practice. |
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04.08.54505 Kamil Kropiewnicki |
Optymalizacja Kombinatoryczna Impossibility of Distributed Consensus with One Faulty Proces |
he consensus problem involves an asynchronous system of processes, some of which may be unreliable. The problem is for reliable processes to agree on a binary value. In this paper, it is shown that every protocol for this problem has the possibility of nontermination, even with only one faulty process. By way of contrast, solutions are known for the synchronous case, the “Byzantine Generals” problem. Authors of the paper were awarded a Dijkstra Prize for this work - given to the most influential papers in distributed computing. |
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26.04.54482 Filip Bartodziej |
Optymalizacja Kombinatoryczna How to eat 4/9 of a pizza |
Unevenly cut pizza is a frustrating occurrence. How can we then make sure that a friend is not trying to reduce our portion of the delicious meal? We will present a strategy which guarantees that one will leave the table satisfied, assuming that they started eating first. Kolja Knauer, Piotr Micek, Torsten Ueckerdt. How to eat 4/9 of a pizza. arXiv. 2008. |
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23.11.51634 Bartosz Podkanowicz |
Podstawy Informatyki Riordan arrays and combinatorial sums by Renzo Sprugnoli |
| The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems. |
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29.03.35340 Krzysztof Michalik |
Optymalizacja Kombinatoryczna Coloring planar graphs with 3 colors and no large monochromatic components |
I will present a proof that there exists a function f(d), such that there exists a 3-coloring of any planar graph G in which each monochromatic subgraph has at most f(d) vertices, where d is the degree of the highest-degree vertex in G. |
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20.12.35316 Mateusz Kaczmarek |
Optymalizacja Kombinatoryczna Hadwiger’s conjecture |
Survey of Hadwiger's Conjecture from 1943, that for all t ≥ 0, every graph is either t-colorable or has a subgraph that can be contracted to the complete t+1 vertices graph. This conjecture is the tremendous strengthening of the four-color problem also known as map coloring problem. |
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18.03.35262 Krzysztof Pióro, Krzysztof Potępa |
Linear-Space Data Structures for Range Mode Query in Arrays [Chan, Durocher, Larsen, Morrison, Wilkinson] |
| Modą multizbioru S nazywamy element, który występuje w S najczęściej, tzn. występuje w S co najmniej tyle razy co każdy inny element S. Mając daną n-elementową tablicę A[1:n] rozważamy prosty problem: konstrukcję statycznej struktury danych pozwalającej szybko odpowiadać na zapytania o modę na przedziale A. Każde zapytanie składa się z pary (i,j), dla której odpowiedzią jest moda A[i:j]. Autorzy pracy prezentują strukturę danych z liniową pamięcią odpowiadającą na zapytania w czasie O(sqrt(n / log n)). Dodatkowo pokazują silną przesłankę, że czas zapytania zdecydowanie niższy od sqrt(n) nie może być uzyskany przy użyciu czysto kombinatorycznych technik - mnożenie macierzy logicznych rozmiaru sqrt(n) x sqrt(n) redukuje się do n zapytań o modę na przedziale w tablicy rozmiaru O(n). Autorzy prezentują też struktury danych dla ortogonalnych zapytań w wyższych wymiarach (zapytania w czasie bliskim O(n1-1/2d)) oraz zapytań o półprzestrzenie (zapytania w czasie O(n1-1/d^2)). |
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23.01.32579 Adam Polak |
Informatyka Teoretyczna Monochromatic triangles, intermediate matrix products, and convolutions |
The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(nω) time algorithms, for ω<2.373. On the other hand, the (min,+)-product, which is at the heart of APSP, is widely conjectured to require cubic time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems, e.g. the (min,max)-product, the min-witness product, APSP in directed unweighted graphs. The best runtimes for these "intermediate" problems are all O(n(3+ω)/2). A similar phenomenon occurs for convolution problems.
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18.07.32469 Mateusz Górski |
Podstawy Informatyki A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic by Paul Wild, Lutz Schroder, Dirk Pattinson and Barbara Konig. |
| The fuzzy modality probably is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that every formula in probabilistic fuzzy first-order logic that is non-expansive wrt. behavioural distance can be approximated by concepts of bounded rank in probabilistic fuzzy description logic. |
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15.08.16151 Kornel Dulęba |
Optymalizacja Kombinatoryczna The Return of Coppersmith’s Atack: Practical Factorization of Widely Used RSA Moduli |
During the seminar I will discuss a clever attack on RSA library used in Infineon chips. Researchers discovered that the prime factors used for constructing private keys have a peculiar form. This allowed them to use a modified version of Coppersmith algorithm to recover private key basing on their public counterpart in a reasonable time for keys up to 2048 bit long. |
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13.03.13304 Michał Zwonek |
Podstawy Informatyki Probably Half True: Probabilistic Satisfability over Lukasiewicz Infnitely-valued Logic by Marcelo Finger and Sandro Preto |
| We study probabilistic-logic reasoning in a context that allows for "partial truths", focusing on computational and algorithmic properties of non-classical Lukasiewicz In nitely-valued Probabilistic Logic. In particular, we study the satis ability of joint probabilistic assignments, which we call LIPSAT. Although the search space is initially in nite, we provide linear algebraic methods that guarantee polynomial size witnesses, placing LIPSAT complexity in the NP-complete class. An exact satis ability decision algorithm is presented which employs, as a subroutine, the decision problem for Lukasiewicz In nitely-valued (non probabilistic) logic, that is also an NP-complete problem. We develop implementations of the algorithms described and discuss the empirical presence of a phase transition behavior for those implementations. |
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27.05.79123 Mikołaj Twaróg |
Optymalizacja Kombinatoryczna A Short Guide to Hackenbush |
Hackenbush is a two player game played on a graph with a few marked vertices. Players alternate turns. Each turn consists of removing one edge from the graph and all vertices that lost their connection to all marked ones. Player, that can't make a move, loses. I will present three different variants of Hackenbush(Red-Blue Hackenbush, Green Hackenbush and R-G-B Hackenbush) with methods to determine, which player has a winning strategy. Padraic Bartlett. A Short Guide to Hackenbush. VIGRE REU 2006. |
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22.08.79068 Katarzyna Bułat, Dawid Tracz |
Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time [P. Parys] |
Gry parzystości to gry pomiędzy dwoma graczami (zwyczajowo Even oraz Odd) na grafie skierowanym G = (V, E). Gracze przesuwają między wierzchołkami wirtualny token, tworząc ścieżkę. Wierzchołki grafu są etykietowane liczbami naturalnymi i każdy z nich jest przypisany do jednego gracza, który decyduje w jakim kierunku zostanie wykonany ruch z tego wierzchołka. Celem każdego z graczy jest wybranie takiej strategii, że przy nieskończonej grze (ścieżce), najwyższa nieskończenie wiele razy powtarzająca się etykieta będzie odpowiednio parzysta bądź nieparzysta. Problem gry parzystości jest deterministyczny, to znaczy dla każdego wierzchołka jeden z graczy posiada strategię wygrywającą. Rekurencyjny algorytm Zielonki rozwiązuje grę parzystości w czasie wykładniczym. Istnieje jednak algorytm działający w czasie quasi-wielomianowym, czyli 2O((log(n))^c) dla pewnego, ustalonego c. W trakcie prezentacji omówiony zostanie schemat nowej wersji algorytmu, przeprowadzona analiza jego złożoności oraz przedstawiony dowód poprawności zwracanego przez niego wyniku. |
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29.06.76385 22.02.57220 Patryk Mikos |
Informatyka Teoretyczna Efficient enumeration of non-isomorphic interval graphs |
Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on n vertices with an O(n4) time delay between the output of two consecutive graphs. We improve their algorithm and achieve O(n3 log n) time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all n up to 15 (previous best was 12). |
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23.12.76275 Piotr Mikołajczyk |
Podstawy Informatyki Satisfiability in Strategy Logic can be Easier than Model Checking by Erman Acar, Massimo Benerecetti and Fabio Mogavero. |
In the design of complex systems, model-checking and satisfiability arise as two prominent decision problems. While The SL fragment we consider is obtained by preventing strategic quantifications within the scope of temporal operators. The resulting logic is quite powerful, still allowing to express important game-theoretic properties of multi-agent systems, such as existence of Nash and immune equilibria, as well as to formalize the rational synthesis problem. We show that satisfiability for such a fragment is PSPACE-COMPLETE, while its model-checking complexity is 2EXPTIME-HARD. The result is obtained by means of an elegant encoding of the problem into the satisfiability of conjunctive-binding first-order logic, a recently discovered decidable fragment of first-order logic. |
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28.04.59981 Adrian Siwiec |
Optymalizacja Kombinatoryczna Edge Coloring Signed Graphs |
We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. It then turns out that Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs. |
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19.01.59958 Paweł Palenica |
Optymalizacja Kombinatoryczna Guess the Larger Number |
We will discuss variations of a zero sum game where one player writes down two numbers on cards. The second player learns one of the numbers to make a guess which of the numbers is larger. We will show variations of the game where the second player has a greater chance of winning than 1/2. |
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18.04.59903 Jędrzej Kula, Przemysław Simajchel |
Subtree Isomorphism Revisited [A. Abboud et al.] |
Problem Izomorfizmu Poddrzew zadaje pytanie, czy dane drzewo zawarte jest w innym danym drzewie. Ten problem o zasadniczym znaczeniu dla algorytmiki jest badany już od lat 60. ubiegłego wieku. Podkwadratowe algorytmy znane są dla niektórych wariantów, np. drzew uporządkowanych, ale nie w ogólnym przypadku. Poprzez redukcję z problemu Wektorów Ortogonalnych pokażemy, że prawdziwie podkwadratowy algorytm dla Izomorfizmu Poddrzew przeczy SETH. Dodatkowo pokażemy, że to samo ograniczenie dolne utrzymuje się również w przypadku ukorzenionych drzew o logarytmicznej wysokości. W opozycji do niego zaprezentujemy również podkwadratowy algorytm randomizowany dla drzew o stałym stopniu i logarytmicznej wysokości. Redukcja korzysta z nowych "gadżetów drzewowych", które prawdopodobnie przydadzą się w przyszłości w wyznaczaniu ograniczeń dolnych opartych na SETH dla problemów na drzewach. Algorytmy opierają się na znanych wynikach o złożoności randomizowanych drzew decyzyjnych. |
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18.08.57110 Edyta Garbarz |
Podstawy Informatyki Unifying Logical and Statistical AI Pedro by Domingos, Daniel Lowd, Stanley Kok, Aniruddh Nath, Hoifung Poon Matthew Richardson and Parag Singla |
| Intelligent agents must be able to handle the complexity and uncertainty of the real world. Logical AI has focused mainly on the former, and statistical AI on the latter. Markov logic combines the two by attaching weights to first-order formulas and viewing them as templates for features of Markov networks. Inference algorithms for Markov logic draw on ideas from satisfiability, Markov chain Monte Carlo and knowledge-based model construction. Learning algorithms are based on the voted perceptron, pseudo-likelihood and inductive logic programming. Markov logic has been successfully applied to a wide variety of problems in natural language understanding, vision, computational biology, social networks and others, and is the basis of the open-source Alchemy system. |
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17.10.38054 Grzegorz Guśpiel |
Informatyka Teoretyczna Smaller Universal Targets for Homomorphisms of Edge-Colored Graphs |
The density D(G) of a graph G is the maximum ratio of the number of edges to the number of vertices ranging over all subgraphs of G. For a class F of graphs, the value D(F) is the supremum of densities of graphs in F. A k-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,...,k. A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class F of graphs, a k-edge-colored graph H (not necessarily with the underlying graph in F) is k-universal for F when any k-edge-colored graph with the underlying graph in F admits a homomorphism to H. Such graphs are known to exist exactly for classes F of graphs with acyclic chromatic number bounded by a constant. The minimum number of vertices in a k-uniform graph for a class F is known to be Ω(kD(F)) and O(kd), where d is the ceiling of D(F) (result obtained in 2015 with Gutowski), and has been conjectured to be ϴ(kD(F)). In this talk, I will present a construction of a k-universal graph on O(kd) vertices for any rational bound d on the density D(F). It follows that if D(F) is rational, the minimum number of vertices in a k-universal graph for F is indeed ϴ(kD(F)). |
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12.04.37945 Jan Kościsz |
Podstawy Informatyki Bohm's Theorem, Church's Delta, Numeral Systems, and Ershov Morphisms by Richard Statman and Henk Barendregt |
| In this note we work with untyped lambda terms under beta-conversion and consider the possibility of extending Bohm's theorem to infnite RE (recursively enumerable) sets. Bohm's theorem fails in general for such sets V even if it holds for all fnite subsets of it. It turns out that generalizing Bohm's theorem to infnite sets involves three other superfcially unrelated notions; namely, Church's delta, numeral systems, and Ershov morphisms. Our principal result is that Bohm's theorem holds for an infnite RE set V closed under beta conversion iff V can be endowed with the structure of a numeral system with predecessor iff there is a Church delta (conditional) for V iff every Ershov morphism with domain V can be represented by a lambda term |
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09.05.21627 Kamil Rajtar |
Optymalizacja Kombinatoryczna A Price-Based Iterative Double Auction for Charger Sharing Markets |
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05.08.21572 Nicoll Bryła, Mateusz Pabian |
Faster Algorithms for All Pairs Non-Decreasing Paths Problem [Duan, Jin, Wu] |
| W tej pracy autorzy prezentują poprawiony algorytm dla problemu wszystkich par ścieżek niemalejących (APNP) dla grafów prostych, skierowanych i ważonych o czasie działania Õ(n^((3+ω)/2)) = Õ(n^2,686), gdzie n jest liczbą wierchołków, a ω jest wykładnikiem złożoności algorytmu szybkiego mnożenia macierzy z pracy [Williams 2012, Le Gall 2014]. To odpowiada najlepszemu, obecnemu górnemu ograniczeniu dla (max, min)-iloczynu macierzy, który można zredukować do APNP. Następne usprawnienia dla APNP implikują szybszy algorytm dla (max, min)-iloczynu macierzy. Poprzednie najlepsze oszacowanie górne dla ważonych, skierowanych grafów było Õ(n^(1/2(3+(3-ω)/(ω+1) + ω))) = Õ(n^2,78) [Duan, Gu, Zhang 2018]. Autorzy pokazują również algorytm Õ(n^2) dla APNP w nieskierowanych, prostych grafach, który również osiąga optimum z czynnikiem logarytmicznym. |
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06.12.18779 Rafał Burczyński |
Podstawy Informatyki Compaction of Church Numerals by Isamu Furuya and Takuya Kida |
| In this study, we address the problem of compaction of Church numerals. Church numerals are unary representations of natural numbers on the scheme of lambda terms. We propose a novel decomposition scheme from a given natural number into an arithmetic expression using tetration, which enables us to obtain a compact representation of lambda terms that leads to the Church numeral of the natural number. For natural number n, we prove that the size of the lambda term obtained by the proposed method is O((s log2n)^(log n/ (log log n))). Moreover, we experimentally confirmed that the proposed method outperforms binary representation of Church numerals on average, when n is less than approximately 10,000 . |
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16.02.84599 Bartosz Walczak |
Informatyka Teoretyczna Coloring and Maximum Weight Independent Set of rectangles |
We prove that every intersection graph of axis-parallel rectangles in the plane with clique number ω has chromatic number Joint work with Parinya Chalermsook. |
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11.08.84489 Jan Mełech |
Podstawy Informatyki On compressing and indexing repetitive sequences by Sebastian Kreft and Gonzalo Navarro |
| We introduce LZ-End, a new member of the Lempel–Ziv family of text compressors, which achieves compression ratios close to those of LZ77 but is much faster at extracting arbitrary text substrings. We then build the first self-index based on LZ77 (or LZ-End) compression, which in addition to text extraction offers fast indexed searches on the compressed text. This self-index is particularly effective for representing highly repetitive sequence collections, which arise for example when storing versioned documents, software repositories, periodic publications, and biological sequence databases. |
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08.09.68171 Vladyslav Rachek |
Optymalizacja Kombinatoryczna On Chromatic Number of Exact Distance Graphs |
For any graph G and positive integer p we can consider "exact distance graph" G in which vertices x and y are connected if and only if their distance in G is exactly p. We can bound chromatic number of such graphs using notion of weak coloring numbers. Proof becomes particularly valuable for odd p and planar graphs G. |
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12.10.65433 Gwenaël Joret Université Libre de Bruxelles |
Informatyka Teoretyczna A new proof of the Erdős-Pósa theorem with applications |
A classic result of Erdős and Pósa (1965) states that for every graph G and every integer k, either G has k vertex-disjoint cycles, or G has a set of Joint work with Henning Bruhn, Wouter Cames van Batenburg, and Arthur Ulmer. |
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06.04.65324 Rafał Byczek |
Podstawy Informatyki Suffix array and Lyndon factorization of a text by Sabrina Mantaci, Antonio Restivo, Giovanna Rosone and Marinella Sciortino |
| The main goal ofthis paper is to highlight the relationship between the suffix array of a text and its Lyndon factorization. It is proved in [15]that one can obtain the Lyndon factorization of a text from its suffix array. Conversely, here we show a new method for constructing the suffix array of a text that takes advantage of its Lyndon factorization. The surprising consequence of our results is that, in order to construct the suffix array, the local suffixes inside each Lyndon factor can be separately processed, allowing different implementative scenarios, such as online, external and internal memory, or parallel implementations. Based on our results, the algorithm that we propose sorts the suffixes by starting from the leftmost Lyndon factors, even if the whole text or the complete Lyndon factorization are not yet available. |
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04.05.49006 Vladyslav Rachek |
Optymalizacja Kombinatoryczna Steinberg's conjecture is false |
It's commonly known that planar graphs are at most 4-colorable. One possible direction towards determining when planar graphs can be 3-colorable is to narrow to particular planar graphs with enforced structure. For example, one can forbid cycles of length 4,5,...,k where k>=4. There is a conjecture of Steinberg from 1976, that planar graphs without cycles of length 4 and 5 (as subgraphs) are 3-colorable. It has been open problem till 2016, when it was disproved in joint paper of Vincent Cohen-Addad, Michael Hebdige, Daniel Kral, Zhentao Li, Esteban Salgado, and we present proof from that paper. |
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31.07.48951 Piotr Helm, Krzysztof Zysiak |
Optimal Sorting with Persistent Comparison Errors [B. Geissmann et al.] |
| Rozważamy problem sortowania n elementów w przypadku stałego błędu porównań. W tym problemie, każde porównanie między dwoma elementami może się pomylić ze stałym (małym) prawdopodobieństwem, i porównania nie mogą zostać powtórzone. Perfekcyjne posortowanie w tym modelu jest niemożliwe i celem jest zminimalizowanie dyslokacji każdego z elementów w zwróconym ciągu, czyli odległość od jego poprawnej pozycji. Istniejące ograniczenia dolne dla tego problemu pokazują, że żaden algorytm nie zagwarantuje z wysokim prawdopodobieństwem maksymalnej i sumarycznej dyslokacji lepszej niż Ω(logn) i Ω(n), odpowiednio, bez względu na czas działania. W tej pracy, prezentujemy pierwszy sortujący algorytm o złożoności O(n log n), który gwarantuje zarówno maksymalna dyslokację O(log n), jak i sumaryczną dyslokację O(n) z wysokim prawdopodobieństwem. To rozstrzyga złożoność czasową tego problemu i pokazuje, że błędy porównań nie zwiększają jego złożoności czasowej: ciąg z najlepszą możliwą dyslokacją może zostać uzyskany w czasie O(n logn), i nawet bez błędów porównań czas Ω(n log n) jest konieczny, by zagwarantować takie ograniczenia dyslokacji. Aby osiągnąć ten optymalny wynik, rozwiązujemy dwa podproblemy, za pomocą metod, które mogą mieć dalsze, osobne zastosowania. Jednym z nich jest zlokalizowanie pozycji, na którą należy wstawić element do prawie posortowanego ciągu o dyslokacji O(log n) w taki sposób, aby dyslokacja zwracanego ciągu wciąż była O(logn). Drugi problem - jak równocześnie wstawić m elementów w prawie posortowany ciąg innych m elementów, tak aby zwracany ciąg 2m elementów pozostał prawie posortowany. |
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06.06.46268 Mikkel Abrahamsen Københavns Universitet |
Informatyka Teoretyczna Geometric Multicut |
We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" F, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as Geometric k-Cut, where k is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an Joint work with Panos Giannopoulos, Maarten Löffler, and Günter Rote. Presented at ICALP 2019. |
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20.10.46158 Maciej Nemś |
Podstawy Informatyki Generating Random Well-Typed Featherweight Java Programs Using Quick Check by Samuel da Silva Feitosaa, Rodrigo Geraldo Ribeirob and Andre Rauber Du Bois |
| Currently, Java is one of the most used programming language, being adopted in many large projects, where applications reach a level of complexity for which manual testing and human inspection are not enough to guarantee quality in software development. Even when using automated unit tests, such tests rarely cover all interesting cases of code, which means that a bug could never be discovered, once the code is tested against the same set of rules over and over again. This paper addresses the problem of generating random well-typed programs in the context of Featherweight Java, a well-known object-oriented calculus, using QuickCheck, a Haskell library for property-based testing. |
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25.03.29786 Bartłomiej Jachowicz, Mateusz Kaczmarek |
Separating strings with small automata [J.M.Robson] |
Tematem pracy jest problem znalezienia automatu skończonego rozróżniającego dwa łańcuchy o możliwie najmniejszej liczbie stanów. Autorzy pokazują, że dla łańcuchów ograniczonych przez długość n istnieje automat akceptujący tylko jeden z łańcuchów o O(n2/5 * log3/5n) stanach, co dla przypadku, gdy łańcuchy na wejściu są równej długości jest najlepszym znanym ograniczeniem.
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25.07.26993 Jacek Kurek |
Podstawy Informatyki GENERIC ALGORITHMS FOR HALTING PROBLEM AND OPTIMAL MACHINES REVISITED |
| The halting problem is undecidable but can it be solved for "most" inputs? This natural question was considered in a number of papers, in diferent settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the notion of Kolmogorov complexity. We also consider some related questions about this framework and about asymptotic properties of the halting problem. In particular, we show that the fraction of terminating programs cannot have a limit, and all limit points are Martin-Lof random reals. We then consider mass problems of finding an approximate solution of halting problem and probabilistic algorithms for them, proving both positive and negative results. We consider the fraction of terminating programs that require a long time for termination, and describe this fraction using the busy beaver function. We also consider approximate versions of separation problems, and revisit Schnorr's results about optimal numberings showing how they can be generalized. |
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22.08.10675 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Choosability number of planar graphs |
During the seminar, we will discuss some open problems regarding the choosability number of planar graphs and related problems. |
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25.10.54481 Bartłomiej Kielak |
Informatyka Teoretyczna Generalized Turán densities and counting cycles in graphs |
The Turán number In this talk, we will show an elementary proof that Joint work with Andrzej Grzesik. |
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24.08.38077 Bruno Pitrus |
Optymalizacja Kombinatoryczna A Borsuk–Ulam Equivalent that Directly Implies Sperner’s Lemma |
It is a known fact that Sperner's purely combinatorial lemma of triangulation is equivalent to theorems in the field of topology: Brouwer with a fixed point and Knastra-Kuratowski-Mazurkiewicz on covering the symplex. Both of these topological theorems have stronger versions (Borsuk-Ulam and Lusternik-Schnirelmann theorems on anti-inflammatory points). In the paper, the authors show a combinatorial analogue of Borsuk-Ulam theorem and use it to directly prove the Sperner lemma, closing the stronger trinity of theorems. |
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17.05.38054 Paweł Palenica |
Optymalizacja Kombinatoryczna Three famous theorems on finite sets |
During the seminar I will present three statements about finite sets with evidence. Two of them are classic theorems of combinatorial power theory - theorems of Sperner and Erdős-Ko-Rado. The third of these is one of the most important theorems in finite set theory - the Hall theorem. |
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20.06.35316 Bartosz Walczak |
Informatyka Teoretyczna Subexponential algorithms for finding large induced sparse subgraphs |
Let 𝒞 and 𝒟 be hereditary graph classes. Consider the following problem: given a graph
This leads, for example, to the following corollaries for specific classes 𝒞 and 𝒟:
Joint work with Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, and Erik Jan van Leeuwen. |
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09.01.18889 Dominika Salawa |
Optymalizacja Kombinatoryczna The Hardness of the Lemmings Game, or Oh no, more NP-Completeness Proofs |
In computer game 'Lemmings', lemmings are placed in a level walking towards certain direction. When they encounter a wall, they turn and walk back in the direction they came from and when they encounter a hole, they fall. If a lemming falls beyond a certain distance, it dies. The goal is to guide lemmings to the exit by assigning them skills and modifying their behavior. I will show by polynomial-time reduction from 3-SAT that deciding whether particular level is solvable is an NP-Complete problem. This holds even if there is only one lemming in the level to save. Graham Cormode. The Hardness of the Lemmings Game, or Oh no, more NP-Completeness Proofs. |
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28.06.16041 Szymon Stankiewicz |
Podstawy Informatyki Bohm's Theorem, Church's Delta, Numeral Systems, and Ershov Morphisms by Richard Statman and Henk Barendregt |
| In this note we work with untyped lambda terms under beta-conversion and consider the possibility of extending Bohm's theorem to in¯nite RE (recursively enumerable) sets. Bohm's theorem fails in general for such sets V even if it holds for all finite subsets of it. It turns out that generalizing Bohm's theorem to infnite sets involves three other superfcially unrelated notions; namely, Church's delta, numeral systems, and Ershov morphisms. Our principal result is that Bohm's theorem holds for an infnite RE set V closed under beta conversion iff V can be endowed with the structure of a numeral system withc predecessor iff there is a Church delta (conditional) for V iff every Ershov morphism with domain V can be represented by a lambda term. |
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19.03.13413 Jarosław Grytczuk Politechnika Warszawska |
Algorytmy Randomizowane i Aproksymacyjne Graph polynomials and choosability |
A result of Thomassen asserts that every planar graph is 5-choosable (colorable from arbitrary lists of size 5 preassigned to the vertices of a graph). We prove that every planar graph has a matching whose deletion gives a 4-choosable graph. The proof is based on Combinatorial Nullstellensatz - a famous algebraic result of Alon involving multivariable polynomials. We also discuss possible applications of this method to other graph coloring problems, like the four color problem or the empire coloring problem, for instance.
Joint work with Xuding Zhu. |
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05.03.81751 Bartłomiej Puget |
Podstawy Informatyki Solving the Rubik’s Cube Optimally is NP-complete by Erik D. Demaine and Sarah Eisenstat |
| In this paper, we prove that optimally solving an n × n × n Rubik’s Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an n×n×n Rubik’s Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik’s Square – an n × n × 1 generalization of the Rubik’s Cube – and then proceed with a similar but more complicated proof for the Rubik’s Cube case. Our results hold both when the goal is make the sides monochromatic and when the goal is to put each sticker into a specific location. |
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28.10.62585 Maciej Czerwiński |
Podstawy Informatyki Automata Theoretic Account of Proof Search by Aleksy Schubert, Wil Dekkers and Henk P. Barendregt |
| Techniques from automata theory are developed that handle search for inhabitants in the simply typed lambda calculus. The resulting method for inhabitant search, which can be viewed as proof search by the Curry-Howard isomorphism, is proven to be adequate by a reduction of the inhabitant existence problem to the emptiness problem for appropriately defined automata. To strengthen the claim, it is demonstrated that the latter has the same complexity as the former. We also discuss the basic closure properties of the automata. |
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05.01.46268 Krzysztof Maziarz |
Optymalizacja Kombinatoryczna Exact Algorithms via Monotone Local Search |
Parameterized algorithms can solve some optimization problems quickly, assuming a certain parameter is bounded: for example, when we aim to satisfy a SAT formula by setting at most k (out of n) variables to true. However, the same algorithms directly applied to the unbounded case (k = n) usually yield poor results. Here I will discuss a bridge between parameterized algorithms and general exact exponential-time algorithms. I will show a remarkably simple approach to obtaining a good exact exponential-time algorithm, given a good parameterized algorithm. The resulting algorithm will be randomized, but it is also possible to derandomize it with subexponential additional cost in the complexity. This approach, at the time of publishing, pushed the state-of-the-art for many optimization problems. |
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08.02.43530 Krzysztof Kleiner |
Informatyka Teoretyczna Range queries and counting triangles |
Listing and counting triangles in sparse graphs are well-studied problems. For a graph with m edges, Björklund et al. gave an O(m1.408) algorithm which can list up to m triangles. The exact exponent depends on the exponent omega in matrix multiplication, and becomes 4/3 if omega=2. Pătraşcu proved that an algorithm faster than O(m4/3) would imply a sub-quadratic algorithm for 3-SUM, which is considered unlikely. In our work we consider a variant of triangle problem asking to determine for every edge the number of triangles which contains that edge. We prove that this problem is no easier than listing up to m triangles, although it still admits an algorithm of the same O(m1.408) complexity. We also propose a natural class of range query problems, including for example the following problem: given a family of ranges in an array, compute the number of inversions in each of them. We prove that all the problems in this class are equivalent, under one-to-polylog reductions, to counting triangles for each edge. In particular the time complexities of these problems are the same up to polylogarithmic factors. This is joint work of Lech Duraj, Krzysztof Kleiner, Adam Polak and Virginia Vassilevska-Williams. |
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23.06.43420 Przemysław Rutka (Lublin) |
Podstawy Informatyki Wybrane algorytmiczne zastosowania klasycznych wielomianów ortogonalnych |
Klasyczne wielomiany ortogonalne, odpowiadające im klasyczne funkcje wagowe oraz ich własności znajdują wiele zastosowań w takich chociażby obszarach jak tomografia, mechanika kwantowa, kombinatoryka, przetwarzanie obrazów i sygnałów, kompresja danych oraz zwiększanie wydajności algorytmów. W tym ostatnim zakresie cały czas uzyskuje się wiele ciekawych wyników, pozwalających na efektywne numeryczne rozwiązywanie różnych problemów. Można do tych problemów w szczególności zaliczyć barycentryczne interpolacje Fejéra, Hermite'a i Lagrange'a oraz problemy ekstremalne typu Szegő i Markowa-Bernsteina. W pierwszym przypadku, gdy interpolowanych jest n wartości w węzłach, będących zerami klasycznych wielomianów ortogonalnych, możliwa jest poprawa złożoności obliczeniowej algorytmów, obliczających wartości wielomianów interpolacyjnych w oparciu o wzory barycentryczne, z O(n^2) do O(n). Wymagane jest w tym celu zastosowanie odpowiednich jawnych wzorów na wagi barycentryczne lub wzorów wiążących wagi barycentryczne z wagami i węzłami kwadratur Gaussa. Z kolei w drugim przypadku, jak się okazuje powiązanym z pierwszym, daje się sformułować wzory, pozwalające bezpośrednio obliczać na komputerze najlepsze stałe, występujące w nierównościach typu Szegő i Markowa-Bernsteina oraz wartości wielomianów ekstremalnych, dla których te nierówności stają się równościami. Nierówności te związane są z iterowanymi klasycznymi funkcjami wagowymi i można je wykorzystać do szacowania wartości lub norm pochodnych D^{k}p lub różnic progresywnych Δ^{k}p wielomianów p(x), odpowiednio w przypadku ciągłym lub dyskretnym.
Inne tego typu rezultaty, korzystające z klasycznych wag i/lub klasycznych wielomianów ortogonalnych, można otrzymać także dla problemu typu izoperymetrycznego w klasie płaskich, zamkniętych krzywych wielomianowych, problemu równowagi elektrostatycznej układu ładunków, problemu efektywnej, stabilnej i najbardziej ekonomicznej interpolacji oraz problemu dwustronnych oszacowań aproksymacyjnych a priori typu Chernoffa. |
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08.12.27125 Anita Badyl |
Optymalizacja Kombinatoryczna A Simplification of the MV Matching Algorithm and its Proof |
Simple and effective algorithms solving the problem of finding maximum matchings in bipartite graphs had been known for years before a low-complexity algorithm for non-bipartite graphs was published for the first time. That algorithm is known as the Micali-Vazirani algorithm, and it constitutes an intricate combination of the Hopcroft-Karp algorithm for bipartite graphs and the Blossom algorithm for general graphs. It achieves the complexity of O(m√n), which demonstrates that matchings in general graphs are not harder to find than matchings in bipartite ones. We present an intuitive introduction to the algorithm, explaining its main definitions and procedures. Vijay V. Vazirani. A Simplification of the MV Matching Algorithm and its Proof. arXiv. 2012. |
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31.08.27102 Kamil Rajtar |
Optymalizacja Kombinatoryczna Rectangular tiling |
During the seminar will be presented proofs of the seemingly geometrical problem of tiling a rectangle with tiles with at least one side of total length. |
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16.02.24255 Weronika Grzybowska |
Podstawy Informatyki A Mesh of Automata by Sabine Broda, Markus Holzer, Eva Maia, Nelma Moreira, Rogerio Reis |
| We contribute new relations to the taxonomy of di erent conversions from regular expressions to equivalent nite automata. In particular, we are interested in transformations that construct automata such as, the follow automaton, the partial derivative automaton, the prefix automaton, the automata based on pointed expressions recently introduced and studied, and last but not least the position, or Glushkov automaton (A_POS), and their double reversed construction counterparts. We deepen the understanding of these constructions and show that with the artefacts used to construct the Glushkov automaton one is able to capture most of them. As a byproduct we define a dual version of the position automaton which plays a similar role as A_POS but now for the reverse expression. Moreover, it turns out that the prefix automaton A_Pre is central to reverse expressions, because the determinisation of the double reversal of A_Pre (first reverse the expression, construct the automaton A_Pre, and then reverse the automaton) can be represented as a quotient of any of the considered deterministic automata that we consider in this investigation. This shows that although the conversion of regular expressions and reversal of regular expressions to nite automata seems quite similar, there are signifcant differences. |
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13.05.70932 Michał Stobierski |
Optymalizacja Kombinatoryczna How 'hard' a video game can be? |
Computer games are a well-studied branch of the theory of complexity. Many of them fit into a similar scheme, lying in the NP (and even NP-hard) and, thanks to Savitch's Theorem, in PSPACE (-hard). It turns out, however, that some of them, thanks to their unique mechanics, are able to simulate the operation of the Turing Machine and thus pose undecidable problems! An interesting example of such a game is Braid, on which this presentation is based. We will start by showing differences and similarities with other games, then we will show how to simulate the operation of the abstract 'counter machine' and talk about a particularly interesting variant of the game, which introduces an TM model that, when it writes to the tape, deletes all data on the tape to the right of the head. And despite the fact that it looks like simplified variant, it lies in EXPSPACE, making Braid a totally 'non-schematic' game. |
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03.02.70909 Rafał Byczek |
Optymalizacja Kombinatoryczna The chromatic number of Kneser graphs |
In 1955 the number theorist Martin Kneser posed a seemingly innocuous problem that became one of the great challenges in graph theory until a brilliant and totally unexpected solution, using the “Borsuk–Ulam theorem” from topology, was found by László Lovász twenty-three years later. It happens often in mathematics that once a proof for a long-standing problem is found, a shorter one quickly follows, and so it was in this case. Within weeks Imre Bárány showed how to combine the Borsuk–Ulam theorem with another known result to elegantly settle Kneser’s conjecture. Then in 2002 Joshua Greene, an undergraduate student, simplified Bárány’s argument even further, and it is his version of the proof that I present here. |
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09.03.68171 Bartłomiej Bosek |
Informatyka Teoretyczna Algorithms for posets and graphs games – coloring and matching |
Graph colorings and online algorithms on graphs constitute the key fragments of the algorithmic graph theory. Specifically, the subject of this study will be a presentation of the results concerning
The first part of the talk will concern different aspects of the coloring problem as well as different evidential techniques. The presented results concern majority choosability of digraphs, harmonious coloring of hypergraphs and semi-uni conjecture of product of two posets. The next part of presentation will concern online chain partitioning of posets. There will be presented a full characterization of the class of posets, for which the number of colors (chains) used by first-fit is a function of width, i.e. best offline solution. This part will also present two different subexponential online algorithm for the online chain partitioning problem. The last part will concern the incremental matching problem in bipartite graphs. There will be presented an incremental algorithm that maintains the maximum size matching in total time equal the running time of one of the fastest offline maximum matching algorithm that was given by Hopcroft and Karp. Moreover, I will show an analysis of the shortest augmenting path algorithm. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Sebastian Czerwiński, Paweł Rzążewski, Stefan Felsner, Kolja Knauer, Grzegorz Matecki, Tomasz Krawczyk, H. A. Kierstead, Matthew Smith, Dariusz Leniowski, Piotr Sankowski, Anna Zych-Pawlewicz. |
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27.04.68112 Bartłomiej Jachowicz, Mateusz Kaczmarek |
On the Complexity of Exact Pattern Matching in Graphs: Binary Strings and Bounded Degree (M. Equi et al.) |
| Szukanie dokładnego wzorca w grafie etykietowanym to problem polegający na szukaniu ścieżek w grafie G = (V, E), których etykiety tworzą napis taki sam jak wzorzec P[1…m]. Ten problem można rozwiązać za pomocą algorytmu działającego w kwadratowym czasie O(|E|m). Jednakże w tej pracy, autorzy podają warunkowe ograniczenie dolne na czas działania algorytmu. Przy założeniu Strong Exponential Time Hypothesis (SETH) nie istnieje algorytm działający w czasie O(m |E|1-e) lub O(|E| m1-e) dla dowolnej stałej e > 0. |
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02.11.49005 27.06.29840 Tomasz Krawczyk |
Informatyka Teoretyczna Testing isomorphism of circular-arc graphs - Hsu's approach revisited |
Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in a circular-arc graph and can be seen as its canonical representations; in particular, every intersection model can be easily transformed into a normalized one.
Our work adapts and appropriately extends the previous work on similar topic done by Hsu [SIAM J. Comput. 24(3), 411--439, (1995)]. In his work Hsu developed decomposition trees representing the structure of all normalized models of circular-arc graphs. However, due to the counterexample given in [Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013] his decomposition trees can not be used by the algorithm testing isomorphism of circular-arc graphs. |
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21.12.48946 Rafał Kaszuba, Michał Zwonek |
A simpler implementation and analysis of Chazelle’s Soft Heaps (H. Kaplan, U. Zwick) |
| W 2000 roku Chazelle wymyślił nową strukturę danych: aproksymacyjne priorytetowe kolejki złączalne (Soft Heaps) i użył jej aby uzyskać najszybszy znany deterministyczny algorytm oparty na porównaniach do obliczenia minimalnego drzewa rozpinającego, jak również nowe algorytmy do znajdowania k-tej najmniejszej liczby na liście i przybliżonego sortowania. Jeśli wstawimy do kolekcji miękkich kopców n elementów to co najwyżej εn ze wszystkich elementów będących aktualnie w kopcach dla danego parametru ε może być uszkodzonych, to znaczy ich klucze zostały sztucznie podwyższone. Dzięki pozwoleniu na uszkodzenia każda operacja na miękkim kopcu jest wykonywana w O(log 1/ε) amortyzowanym czasie. Chazelle uzyskał miękkie kopce przy pomocy kopców dwumianowych, gdzie każda kolejka priorytetowa to kolekcja drzew dwumianowych. W tej pracy autorzy opisują prostszą i bardziej bezpośrednią implementację miękkich kopców, gdzie każda kolejka priorytetowa jest złożona z kolekcji standardowych drzew binarnych. Ta implementacja ma przewagę nad wcześniejszą, bo nie trzeba wykonywać operacji sprzątania, której używał Chazelle w swojej. W pracy przedstawiona jest również zwięzła analiza amortyzowana nowej implementacji. |
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16.03.48896 Dawid Tracz |
Podstawy Informatyki Regular Matching and Inclusion on Compressed Tree Patterns with Context Variables by Iovka Boneva, Joachim Niehren, and Momar Sakho |
| We study the complexity of regular matching and inclusion for compressed tree patterns extended by context variables. The addition of context variables to tree patterns permits us to properly capture compressed string patterns but also compressed patterns for unranked trees with tree and hedge variables. Regular inclusion for the latter is relevant to certain query answering on Xml streams with references. |
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01.09.32601 Filip Bartodziej |
Optymalizacja Kombinatoryczna Turán’s graph theorem |
We’ll cover the Turan theorem from 1941, which provides a restriction on the number of edges in a graph that doesn’t contain an induced k-clique, depending on parameter k. |
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24.05.32578 Mateusz Pabian |
Optymalizacja Kombinatoryczna Gaming is a hard job, but someone has to do it! |
General schemes relating the computational complex-ity of a video game to the presence of certain common elements or mechan-ics, such as destroyable paths, collectible items, doors opened by keys or activated by buttons or pressure plates, etc. Proofs of complexity of several video games, including Pac-Man, Tron, Lode Runner, Boulder Dash, Deflektor, Mindbender, Pipe Mania, Skweek, Prince of Persia, Lemmings, Doom, Puzzle Bobble 3, and Starcraft. Giovanni Viglietta. Gaming is a hard job, but someone has to do it! arXiv. 2013. |
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10.11.29730 Jan Derbisz |
Podstawy Informatyki What Percentage of Programs Halt? by Laurent Laurent Bienvenu, Damien Desfontaines and Alexander Shen |
| Fix an optimal Turing machine U and for each n consider the ratio \rho^U_n of the number of halting programs of length at most n by the total number of such programs. Does this quantity have a limit value? In this paper, we show that it is not the case, and further characterise the reals which can be the limsup of such a sequence \rho^U_n . We also study, for a given optimal machine U, how hard it is to approximate the domain of U from the point of view of coarse and generic computability. |
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26.04.13436 Marcin Briański |
Optymalizacja Kombinatoryczna A short story of graphs that count |
In 1978 Thomason provided a simple, constructive proof of Smith’s theorem; in particular this proof provides a simple algorithm enables one to find a second Hamiltonian cycle whenever one is given a cubic graph and a Hamiltonian cycle in it. For a couple of years, the runtime of the algorithm remained unknown, with worst known cases being cubic (in the number of vertices), however in 1999 Krawczyk found an example of a graph family, such that Thomason’s algorithm takes time Ω(2n/8) where is the number of vertices in the input graph from the family. In this talk, I will present a family of cubic, planar, and 3-connected graphs, such that Thomason’s algorithm takes time Θ(1.1812n) on the graphs in this family. This scaling is currently the best known. |
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01.01.79146 Vladyslav Hlembotskyi |
Optymalizacja Kombinatoryczna The Angel of power 2 wins |
Let's consider the following game: we have two players (they are called the angel and the devil) and an infinite chessboard. The angel is located in some cell on the board. Players make moves alternatively. The devil chooses any cell that is not occupied by the angle and blocks it. The angel can jump to any other cell which is at distance at most p (p is fixed) from its present location and is not blocked. The devil wins if the angel cannot jump to any other cell. The angel wins if it can avoid being captured forever. We will show that the angel of power 2 has a winning strategy. |
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24.09.79122 Katarzyna Bułat |
Optymalizacja Kombinatoryczna Distributed tracing |
The presentation will cover the topic of distributed tracing, which is an important issue in the field of distributed systems. Services are nowadays implemented as complex networks of related sub-systems and it is often hard to determine the source of performance problem in such complex structures. We will take a look at Dapper, a large-scale distributed systems tracing infrastructure, and discuss the challenges its designers had to face, as well as the opportunities the tool gives to programmers. We will discuss the core goals of effective instrumentation, analyze the problem of handling huge amount of tracing data and focus on security concerns. |
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19.05.59957 Adrian Siwiec |
Optymalizacja Kombinatoryczna Online Maximum Matching with Recourse |
Online maximum matching problem has a recourse of k, when the decision whether to accept an edge to a matching can be changed k times, where k is typically a small constant. First, we consider the model in which arriving edge never disapears. We show that greedy algorithm has competitive ratio of 3/2 for even k and 2 for odd k. Then we show an improvement for typical values of k and proceed to show a lower bound of 1+1/(k-1). Later, we discuss a model where edges can appear and disappear at any time and show generalized algorithms. |
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05.11.57109 Rafał Byczek |
Podstawy Informatyki Improving the Upper Bound on the Length of the Shortest Reset Words by Marek Szykula |
| We improve the best known upper bound on the length of the shortest reset words of synchronizing automata. The new bound is slightly better than 114n^3 / 685+O(n^2). The Cerny conjecture states that (n−1)^2 is an upper bound. So far, the best general upper bound was (n^3−n)/6−1 obtained by J.-E. Pin and P. Frankl in 1982. Despite a number of efforts, it remained unchanged for about 35 years. To obtain the new upper bound we utilize avoiding words. A word is avoiding for a state q if after reading the word the automaton cannot be in q. We obtain upper bounds on the length of the shortest avoiding words, and using the approach of Trahtman from 2011 combined with the well-known Frankl theorem from 1982, we improve the general upper bound on the length of the shortest reset words. For all the bounds, there exist polynomial algorithms finding a word of length not exceeding the bound. |
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12.01.40792 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Open problem session |
At the seminar were presented some interesting open problems in the field of graph theory. |
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14.05.37999 Kornel Dulęba, Jan Mełech |
A Randomized Maximum-Flow Algorithm (Cheriyan & Hagerup) |
Praca przedstawia randomizowany algorytm obliczający maksymalny przepływ. Dla sieci przepływowej o n wierzchołkach i m krawędziach, czas wykonania jest O(nm + n2(log n)2) z prawdpodobieństwem co najmniej 1 - 2-sqrt(nm). Algorytm jest zawsze poprawny i w najgorszym przypadku działa w czasie O(nm log n). Czynnik randomizujący składa się tylko z zastosowania losowych permutacji do list sąsiedztwa wierzchołków na początku algorytmu. |
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30.06.37944 Vladyslav Hlembotskyi |
Podstawy Informatyki Upper Bounds for Standardizations and an Application by Hongwei Xi |
| We present a new proof for the standardization theorem in lambda-calculus, which is largely built upon a structural induction on lambda-terms. We then extract some bounds for the number of beta-reduction steps in the standard beta-reduction sequence obtained from transforming a given beta-reduction sequence, sharpening the standardization theorem. As an application, we establish a super exponential bound for the lengths of beta-reduction sequences from any given simply typed A |
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06.09.21626 Kamil Kropiewnicki |
Optymalizacja Kombinatoryczna Identities versus bijections |
In 1740 Leonhard Euler began to work on counting partitions. It resulted in two fundamental papers in the field. Integer partitions have been an active field of study ever since, tackled by many including Srinivasa Ramanujan, Paul Erdős and Donald Knuth. We present a few beautiful proofs of identities using only basic generating functions and simple bijections. |
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09.10.18888 Zoltán Lóránt Nagy Eötvös University & Alfréd Rényi Institute of Mathematics |
Informatyka Teoretyczna Triangles in line arrangements |
A widely investigated subject in combinatorial geometry, originating from Erdős, is the following: given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances, e.g., determine the maximum number of unit distances, the maximum number of minimum/maximum distances, the minimum number of distinct distances? This has been generalized in many directions by taking point sets in a certain (not necessarily Euclidean) metric space and studying the distribution of certain configurations — and a whole theory emerged. In this talk I propose the following problem variant: consider planar line arrangements of n lines, and determine the maximum number of unit/maximum/minimum area determined by these lines. We prove that the order of magnitude for the maximum occurrence of unit area lies between Joint work with Gábor Damásdi, Leo Martínez-Sandoval and Dániel T. Nagy. |
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23.02.18779 Jan Derbisz, Pola Kyzioł, Krzysztof Maziarz, Jakub Nowak, Grzegorz Juzrdziński |
Podstawy Informatyki Prezentacje prac magisterskich |
Jan Derbisz, Promotor: dr hab. Tomasz Krawczyk Pola Kyzioł, Promotor: dr hab. Tomasz Krawczyk Krzysztof Maziarz, Promotor: prof. dr hab. Jacek Tabor Jakub Nowak, Promotor: prof. dr hab. Jacek Tabor Grzegorz Jurdziński, Promotor: dr Piotr Micek |
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29.09.76384 Michał Wrona |
Informatyka Teoretyczna Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width |
We study the amount of consistency (measured by relational width) needed to solve the CSP parametrized by first-order expansions of countably infinite homogeneous graphs, that are, the structures first-order-definable in a homogeneous graph containing the edge relation E, the relation N that holds between different vertices not connected by an edge and the equality. We study our problem for structures that additionally have bounded strict width, i.e., establishing local consistency of an instances of the CSP not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. It is known that with every countably infinite homogeneous graph G the finite unique minimal set S of finite graphs is associated such that some finite H is an induced substructure of G if and only if there is no H' in S such that H' embeds into H. |
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02.11.73646 Marcin Briański |
Algorytmy Randomizowane i Aproksymacyjne Measuring sparsity (based on the lecture by M. Pilipczuk and S. Siebertz) |
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09.12.68170 Rafał Burczyński |
Optymalizacja Kombinatoryczna Basic properties of 3CCP graphs |
We will introduce a class of graphs called 3CCP, which contains graphs that are 3-connected, cubic (3-regular) and planar. It was shown by Tarjan that finding Hamiltonian cycle in a graph assuming these properties remains NP-complete - we will show the reduction from 3-SAT problem. After that we will present Smith's theorem about parity of number of Hamiltonian cycles containing given edge in cubic graphs and show elegant constructive proof using Thomason's lollipop method. After that we will show a class of graphs for which previous algorithm for finding second Hamiltonian cycle takes exponential number of steps. |
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28.01.68112 Jan Derbisz, Franciszek Stokowacki |
An Equivalence Class for Orthogonal Vectors (L.Chen, R.Williams) |
| Problem sprawdzania, czy pośród n wektorów istnieje para wektorów ortogonalnych umiemy łatwo rozwiązać w czasie O(n2 log n), jednak nie jest znany algorytm szybszy niż n2. Autorzy pracy dowodzą, że istnienie algorytmu podkwadratowego jest równoważne istnieniu takich algorytmów dla kilku innych problemów, między innymi Apx-Min-IP - znajdowania pary wektorów będących k-aproksymacją maksymalnego iloczynu skalarnego oraz Approximate Bichrom.-ℓp-Closest-Pair - problemu znajdowania aproksymowanej najbliższej dwukolorowej pary punktów. Powyższe równoważności są zachowane w sytuacji, w której zamiast odpowiadać offline mamy strukturę danych i odpowiadamy na zapytania online. Dodatkowo w pracy przedstawione są nowe algorytmy aproksymowane dla Apx-Min-IP oraz rozwiązywania pewnych instancji MAX-SAT. |
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12.01.65433 Lech Duraj |
Informatyka Teoretyczna A sub-quadratic algorithm for Longest Common Increasing Subsequence |
The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated longest common subsequence (LCS). For LCIS, as well as for LCS, there is an O(n2) algorithm and a SETH-based quadratic lower bound. Both the algorithm and the proof of the bound are, however, quite different for LCIS. For LCS, there is also the Masek-Paterson O(n2/log n) algorithm. Its technique (the 'four Russians trick') does not seem to work for LCIS in any obvious way, so a natural question arises: does any sub-quadratic algorithm exist for Longest Common Increasing Subsequence problem? We answer this question positively, presenting a O(n2/logan) algorithm for some a>0. The algorithm is not based on memorizing small inputs (often used for logarithmic speedups, including LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences. |
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04.08.49005 Adrian Siwiec |
Optymalizacja Kombinatoryczna List coloring of Latin Squares |
For each cell (i, j) of NxN square there is given a list C(i, j) of N colors. Can we choose a color for each cell in such a way that colors in each row and each column are distinct? |
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22.09.48946 Katarzyna Bułat, Kamil Rajtar |
Correctness of constructing optimal alphabetic trees reviseted |
Prezentowana przez nas praca przedstawia nowe obserwacje, które pozwoliły autorom dowieść poprawności dwóch znanych algorytmów (Hu-Tuckera i Garsi-Wachs) na konstrukcję optymalnych drzew utrzymujących porządek leksykograficzny. Omówimy uogólnioną wersję algorytmu Garsi-Wachs wraz z przejrzystym i łatwym do zilustrowania dowodem, który pomaga również w zrozumieniu podejścia Hu-Tuckera. |
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07.09.46267 Grzegorz Gutowski |
Informatyka Teoretyczna Entropy Compression for Acylic Edge-Colorings |
Let G be a graph with maximum degree d. We show a randomized procedure that colors the edges of G so that:
Such a coloring is called an acylic edge-coloring of G. The minimum number of colors in an acyclic edge coloring of G is called the acylic index of G. It is conjectured that acylic index of G is at most d+2. We are able to prove that our coloring procedure succeeds for roughly 3.97d colors (improving on a previous result that used 4d colors). This is joint work with Jakub Kozik and Xuding Zhu. |
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22.11.46157 Rafał Byczek i Paweł Mader |
Podstawy Informatyki A theory of linear typings as flows on 3-valent graphs by Noam Zeilberger |
| Building on recently established enumerative connections between lambda calculus and the theory of embedded graphs (or “maps”), this paper develops an analogy between typing (of lambda terms) and coloring (of maps). Our starting point is the classical notion of an abelian group-valued “flow” on an abstract graph (Tutte, 1954). Typing a linear lambda term may be naturally seen as constructing a flow (on an embedded 3-valent graph with boundary) valued in a more general algebraic structure consisting of a preordered set equipped with an “implication” operation and unit satisfying composition, identity, and unit laws. Interesting questions and results from the theory of flows (such as the existence of nowhere-zero flows) may then be re-examined from the standpoint of lambda calculus and logic. For example, we give a characterization of when the local flow relations (across vertices) may be categorically lifted to a global flow relation (across the boundary), proving that this holds just in case the underlying map has the orientation of a lambda term. We also develop a basic theory of rewriting of flows that suggests topological meanings for classical completeness results in combinatory logic, and introduce a polarized notion of flow, which draws connections to the theory of proof-nets in linear logic and to bidirectional typing. |
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11.10.43529 Marcin Briański |
Algorytmy Randomizowane i Aproksymacyjne Measuring sparsity (based on the lecture by M. Pilipczuk and S. Siebertz) |
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29.03.29840 Kamil Kropiewnicki |
Optymalizacja Kombinatoryczna Shuffling cards |
What do the birthday paradox, the coupon collector problem and shuffling cards have in common? What does it mean for a deck of cards to be "random" or "close to random"? How long does one have to shuffle a deck of cards until it is random? In practical use cases, the question is not about the asymptote - it is about the exact numbers. |
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17.05.29781 Bartłomiej Jachowicz, Mateusz Kaczmarek |
SETH-based Lower Bounds for Subset Sum and Bicriteria Path |
| Głównym rezultatem tego artykułu jest ścisła redukcja z k-SAT do problemu Subset Sum na gęstych instancjach, co pokazuje że algorytm Bellmana z 1962 roku O*(T) - dla Subset Sum z n liczbami i celem równym T nie da się poprawić do czasu T1 - e * 2o(n), dla dowolnego e > 0, pod warunkiem prawdziwości SETH. Wnioskiem z tego jest twierdzenie "Direct-OR" dla problemu Subset Sum pod warunkiem prawdziwości SETH, dające nowe możliwości udowadniania dolnych ograniczeń. Daje nam to możliwość założenia, że podjęcie decyzji o tym, czy jedna z N danych instancji problemu Subset Sum jest TAK-instancją wymaga (NT)1-o(1) czasu. Zastosowaniem danego rezultatu jest dolne ograniczenie dla problemu BICRITERIA s,t-PATH pod warunkiem prawdziwośći SETH. |
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17.07.26992 Krzysztof Turowski Purdue University, USA |
Podstawy Informatyki Compression of Dynamic Graphs Generated by a Duplication Model |
One of the important topics in the information theory of non-sequential random data structures such as trees, sets, and graphs is the question of entropy: how many bits on average are needed to describe the structure. Here we consider dynamic graphs generated by a duplication model in which a new vertex selects an existing vertex and copies all of its neighbors. We provide asymptotic formulas for entopies for both labeled and unlabeled versions of such graphs and construct compression algorithms matching these bounds up to two bits. Moreover, as a side result, we were able to derive asymptotic expansions of expected value of f(X) for functions of polynomial growth, when X has beta-binomial distribution - which in turn allowed to obtain e.g. asymptotic formula the entropy for a Dirichlet-multinomial distribution. |
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05.06.24364 Bartosz Wodziński |
Algorytmy Randomizowane i Aproksymacyjne Algorithmic barriers from phase transitions (Dimitris Achlioptas, Amin Coja-Oghlan) |
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22.11.10674 Kamil Rajtar |
Optymalizacja Kombinatoryczna Communication without errors |
Main aim of the lecture is the answer for Claude Shannon's question from 1956: "Suppose we want to transmit messages across a channel (where some symbols may be distorted) to a receiver. What is the maximum rate of transmission such that the receiver may recover the original message without errors?" |
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11.01.10616 Rafał Kaszuba, Krzysztof Zysiak |
Fast Modular Subset Sum using Linear Sketching |
Dostając zbiór n dodatnich liczb całkowitych, problem Modular Subset Sum polega na sprawdzeniu czy istnieje podzbiór, który sumuje się do zadanego t modulo dana liczba całkowita m. Jest to naturalne uogólnienie problemu Subset Sum (m=+∞), który silnie łączy się z addytywną kombinatoryką i kryptografią. Niedawno zostały opracowane efektywne algorytmy dla przypadku niemodularnego, działające w czasie blisko-liniowym pseudo-wielomianowym. Jednak dla przypadku modularnego najlepszy znany algorytm (Koiliaris'a i Xu) działa w czasie Õ(m5/4). W tej pracy prezentujemy algorytm działający w czasie Õ(m), który dopasowuje się do warunkowego ograniczenia dolnego opartego na SETH. W przeciwieństwie do większości poprzednich wyników związanych z problemem Subset Sum, nasz algorytm nie korzysta z FFT. Natomiast, jest zdolny zasymulować "podręcznikowe" programowanie dynamiczne znacznie szybciej, używając pomysłów ze Szkicowania Liniowego. Jest to jedna z pierwszych aplikacji technik bazujących na szkicowaniu, by osiągnąć szybki algorytm dla problemów kombinatorycznych w modelu offline. |
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24.03.57219 Filip Bartodziej |
Optymalizacja Kombinatoryczna Cayley’s formula for the number of trees & How to guard a museum |
First, several proofs for the number of labeled trees, each using different approach (bijection, linear algebra, recursion, double counting) will be presented. Second part of the seminar will introduce an interesting graph problem first raised by Victor Klee in 1973. This problem can be represented as placing guards in a museum to guard it properly - that is area of the museum must be completely covered by the field of view of the guards. |
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26.04.54481 Agnieszka Łupińska University of California, Davis |
Informatyka Teoretyczna Gunrock: GPU Graph Analytics |
Gunrock is a CUDA library for graph-processing designed specifically for the GPU. It uses a high-level, bulk-synchronous, data-centric abstraction focused on operations on a vertex or edge frontier. Gunrock achieves a balance between performance and expressiveness by coupling high performance GPU computing primitives and optimization strategies with a high-level programming model that allows programmers to quickly develop new graph primitives with small code size and minimal GPU programming knowledge. |
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13.07.54371 Jakub Łabaj i Gabriela Czarska |
Podstawy Informatyki Programming Languages Capturing Complexity Classes by LARS KRISTIANSEN and PAUL J. VODA |
| We investigate an imperative and a functional programming language. The computational power of fragments of these languages induce two hierarchies of complexity classes. Our first main theorem says that these hierarchies match, level by level, a complexity-theoretic alternating space-time hierarchy known from the literature. Our second main theorems says that a slightly different complexity-theoretic hierarchy (the Goerdt-Seidl hierarchy) also can be captured by hierarchies induced by fragments of the programming languages. Well known complexity classes like LOGSPACE, LINSPACE, P, PSPACE etc., occur in the hierarchies. |
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31.05.51743 Maciej Czerwiński |
Algorytmy Randomizowane i Aproksymacyjne Lovasz meets Weisfeiler and Leman (by Dell, Grohe and Rattan) |
| "In this paper, we relate a beautiful theory by Lovász with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k -dimensional generalization known as the Weisfeiler-Leman algorithm." |
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23.02.38077 Franciszek Stokowacki |
Optymalizacja Kombinatoryczna An Approximate Restatement of the Four-Color Theorem |
Four color theorem was proven in 1976 with extensive computer help. Since then there is interest in finding a simpler proof that uses no computer computation. I will present relation between Four Color Theorem and edge 3-coloring of planar, cubic graphs without bridges, and a new result proving that the existence of approximate coloring (with the fourth color used ‘rarely’) is enough to imply Four Color Theorem. |
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16.11.38053 Vladyslav Hlembotskyi |
Optymalizacja Kombinatoryczna EERTREE: An Efficient Data Structure for Processing Palindromes in Strings |
A palindrome is a string which reads the same forward as backward, such as `Ada` or `lol`. We will present a data structure which stores information about all the different palindromic substrings of a given string and prove some basic facts about the data structure. We will show that it is useful and discuss some problems which can be solved with it. |
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05.01.37995 Łukasz Miśkiewicz, Adam Pardyl |
Space-Efficient Algorithms for Longest Increasing Subseqence |
| Najdłuższy rosnący podciąg jest znanym problemem, który można rozwiązać w złożoności O(n*log(n)) używając O(n*log(n)) dodatkowych bitów. Autorzy pracy prezentują algorytmy korzystające z mniejszej ilości dodatkowej pamięci. Konkretniej, dla sqrt(n) <= s <= n, pokazują sposób obliczania długości najdłuższego rosnącego podciągu w O(1/s * n2 * log(n)) korzystając z O(s * log(n)) dodatkowych bitów oraz obliczanie tego podciągu w czasie O(1/s * n2 * log2(n)) używając tyle samo dodatkowych bitów. Dodatkowo autorzy dowodzą, że dla danej złożoności pamięciowej złożoności czasowe w modelu dostępu sekwencyjnego są optymalne z dokładnością do czynników polilogarytmicznych. |
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21.12.35315 Łukasz Lachowski |
Informatyka Teoretyczna Complexity of the quorum intersection property of the Federated Byzantine Agreement System |
A Federated Byzantine Agreement System is defined in the paper https://www.stellar.org/
as a pair (V,Q) consisting of a set of nodes V and a quorum function Q : V → P(P(V)) specifying for each node a nonempty family of subsets of nodes, called quorum slices. A subset of nodes is a quorum if and only if for each of its nodes it also contains at least one of its quorum slices. The Disjoint Quorums Problem answers the question whether a given instance of fbas contains two quorums that have no nodes in common. We show that this problem is NP-complete. We also study the problem of finding a quorum of minimal size and show it is NP-hard. Further, we consider the problem of checking whether a given subset of nodes contains a quorum for some selected node. We show this problem is P-complete and describe a method that solves it in linear time with respect to number of nodes and the total size of all quorum slices. Moreover, we analyze the complexity of some of these problems using the parametrized point of view.
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05.05.35206 Dominik Gryboś |
Podstawy Informatyki Characterizing Polynomial and Exponential Complexity Classes in Elementary Lambda-Calculus by Patrick Baillot, Erika De Benedetti, Simona Ronchi Della Rocca |
| In this paper an implicit characterization of the complexity classes k-EXP and k-FEXP, for k \geq 0, is given, by a type assignment system for a stratified lambda - calculus, where types for programs are witnesses of the corresponding complexity class. Types are formulae of Elementary Linear Logic (ELL), and the hierarchy of complexity classes k-EXP is characterized by a hierarchy of types. |
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02.06.18884 Jakub Nowak |
Optymalizacja Kombinatoryczna Snowflake to Avalanche: A Novel Metastable Consensus Protocol Family for Cryptocurrencies |
Consensus is one of the most important goals to be achieved when many distributed computers share the same task and resources. There are two main families of algorithms solving this problem. Traditional consensus protocols require O(n2) communication, while blockchains rely on proof-of-work. In this talk we will introduce a new family of leaderless Byzantine fault tolerance protocols, built on a metastable mechanism. These protocols provide a strong probabilistic safety and are both quiescent and green. We will analyze some of their properties and guarantees. Finally we will see results of porting Bitcoin transactions to the introduced family of protocols. |
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28.12.16040 Bartłomiej Puget |
Podstawy Informatyki THE SAFE LAMBDA CALCULUS by WILLIAM BLUM AND LUKE ONG |
| Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simply-typed lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of beta-reduction that preserves safety. In the same vein as Schwichtenberg’s 1976 characterization of the simply-typed lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the complexity of deciding beta-eta equality of two safe simply-typed terms and show that this problem is PSPACE-hard. Finally we give a game-semantic analysis of safety: We show that safe terms are denoted by P-incrementally justified strategies. Consequently pointers in the game semantics of safe lambda terms are only necessary from order 4 onwards. |
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14.03.81856 Jan Derbisz |
Optymalizacja Kombinatoryczna Choosability of Planar Graphs |
Colorability and choosability of planar graphs have been heavily studied in the past. In 1994 Thomassen proved that every planar graph is 5-choosable using concise induction. Recently Grytczuk and Zhu used similar ideas to prove that for every planar graph G we can find a matching M in it such that G-M is 4-choosable with the help of Combinatorial Nullstellensatz theorem. |
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11.06.81801 Konrad Deka, Szymon Kapała |
Tighter Connections Between Formula-SAT and Shaving Logs |
| W 2015, Abboud, Backurs i Vassilevska-Williams pokazali że algorytm dla LCS działający w czasie O(n2-eps) implikowałby szybki algorytm dla CNF-SAT, i tym samym fałszywość SETH. W tej pracy, na podstawie innych hipotez dotyczących SAT, autorzy szukają dolnych ograniczeń postaci O(n2/logc n) dla LCS, a także problemu odległości Frecheta oraz problemu matchowania regexów. Głównym rezultatem jest redukcja z SAT-a na formule wielkości s, mającej n zmiennych, do LCS na ciągach długości 2n/2s1+o(1). Wynika stąd, że algorytm dla LCS działający w O(n2/log7+epsn) implikowałby fałszywość pewnych hipotez o Formula-SAT, a algorytm działający w O(n2/log17+epsn) - znaczący postęp w teorii złożoności obwodów. |
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27.05.79122 15.08.16150 Piotr Kawałek |
Informatyka Teoretyczna Computational approach to solving equations in finite realms |
Computational approach to the problem of solving equation, began with the question of David Hilbert. He asked, if there exists an algorithm, that decides wheather given Diophantine equation has a solution or not. Yuri Matiyasevich proved this problem to be undecidable. An analogy for decidability in finite realms is tractability. During the talk, we introduce the notion of PolSat problem for finite algebras and discuss the results for the wide class of algebraic structures. |
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09.10.79012 Jacek Kurek i Bruno Pitrus |
Podstawy Informatyki COMPLEXITY PROBLEMS IN ENUMERATIVE COMBINATORICS by IGOR PA |
| The subject of enumerative combinatorics is both classical and modern. It is classical, as the basic counting questions go back millennia; yet it is modern in the use of a large variety of the latest ideas and technical tools from across many areas of mathematics. The remarkable successes from the last few decades have been widely publicized; yet they come at a price, as one wonders if there is anything left to explore. In fact, are there enumerative problems that cannot be resolved with existing technology? In this paper we present many challenges in the field from the computational complexity point of view, and describe how recent results fit into the story. |
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29.06.76384 18.09.13412 Dominika Salawa, Kamil Kropiewnicki |
Algorytmy Randomizowane i Aproksymacyjne Representative sets in matroids (based on chapter of 'Parameterized algorithms') |
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07.11.62690 Krzysztof Maziarz |
Optymalizacja Kombinatoryczna A refinement of choosability of graphs |
Between the well-known concepts of k-colorability and k-choosability (also know as k-list colorability) lies a whole spectrum of more refined notions. This allows for seeing k-colorability and k-choosability under one unified framework. Exploring this, one immediately discovers interesting problems - for example, possible strengthenings of the four color theorem. We will take a look at these notions, prove some of their properties, and leave many conjectures and open problems. |
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04.02.62636 Rafał Byczek, Bruno Pitrus |
Approximating Edit Distance Within Constant Factor in Truly Sub-Quadratic Time |
Odległość edycyjna to jeden ze sposobów zmierzenia jak bardzo dwa ciągi znaków są do siebie podobne. Polega on na zliczeniu minimalnej liczby operacji wstawienia, usunięcia lub zmienienia znaku na inny, wymaganej aby przekształcić jedno słowo w drugie. W tej pracy autorzy skupili się na problemie złożoności obliczeniowej aproksymowania odległości edycyjnej pomiędzy parą słów. Problem wyznaczenia dokładnej odległości edycyjnej może być rozwiązany za pomocą klasycznego algorytmu dynamicznego działającego w kwadratowym czasie. W 2010 roku Andoni, Krauthgamer i Onak przedstawili działający w czasie prawie liniowym, algorytm aproksymujący odległość edycyjną z polilogarytmicznym czynnikiem aproksymacji. W 2014 Backurs i Indyk pokazali, że dokładny algorytm działający w czasie O(n^(2-δ))implikowałby szybki algorytm dla SAT i fałszywość silnej hipotezy o czasie wykładniczym (SETH). Ponadto, ostatnio w 2017, Abboud i Backurs pokazali, że istnienie algorytmu aproksymującego odległość edycyjną w czasie prawdziwie podkwadratowym z czynnikiem aproksymacji 1 + o(1) implikowałoby fałszywość paru hipotez dotyczących złożoności obwodów boolowskich (circuit complexity). To poddaje w wątpliwość aproksymowalność odległości edycyjnej z dokładnością do czynnika stałego w czasie prawdziwie podkwadratowym. W tej pracy autorzy jednak odpowiedzieli twierdząco na to pytanie, przedstawiając bardzo ciekawy algorytm aproksymujący odległość edycyjną, z stałym czynnikiem aproksymacji i dowodząc, że jego czas działania jest ograniczony od góry przez Õ(n^(2−2/7)). |
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04.06.59847 Marcin Briański |
Podstawy Informatyki On the compressibility of finite languages and formal proofs by Sebastian Eberhard and Stefan Hetzl |
| We consider the minimal number of productions needed for a grammar to cover a finite language L as the grammatical complexity of L. We study this measure for several types of word and tree grammars and show that it is closely connected to well-known measures for the complexity of formal proofs in first-order predicate logic. We construct an incompressible sequence of finite word languages and transfer this and several other results about the complexity of word and tree languages to formal proofs |
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22.02.57219 Dawid Tracz |
Algorytmy Randomizowane i Aproksymacyjne Finding Cliques in Social Networks: A New Distribution-Free Model (Fox, Roughgarden, Seshadhri, Wei, Wein) |
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03.07.43525 Jakub Łabaj |
Optymalizacja Kombinatoryczna Contracting a Planar Graph Efficiently |
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Jakub Łabaj. Contracting a Planar Graph Efficiently. slides. 2018. |
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29.09.43470 Tomasz Zieliński, Michał Zwonek |
On the Complexity of the (Approximate) Nearest Colored Node Problem |
| Mając dany graf G = (V, E) gdzie każdy wierzchołek ma przypisany kolor, pytamy o przybliżoną odległość pomiędzy danym wierzchołkiem v a najbliższym jemu kolorowi c. Prezentujemy wyrocznię o rozciągłości 4k-5 wykorzystującą O(kn sigma^(1/k)) przestrzeni i O(log k) czasu zapytania. Następnie dowodzimy, że posiadając estymatę rzędu O(polylog(n)) jesteśmy w stanie w czasie O(1) udzielić odpowiedzi na pytanie o dokładną odległość dist(v, c). Na końcu pokazujemy związek pomiędzy problemem lambda-OuMv a odległością dist(v, c). |
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14.09.40791 19.01.59957 Michał Seweryn |
Informatyka Teoretyczna Bumping a ladder |
We show that every 3-connected graph which contains many disjoint 2xn-grid minors, contains a 2x(n+1)-grid-minor. We use this result in a qualitative structure theorem for graphs without large 2xn grids. This is a result from a joint paper with Tony Huynh, Gwenaël Joret, Piotr Micek and Paul Wollan |
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27.01.40682 Mateusz Tokarz |
Podstawy Informatyki Enumerating Proofs of Positive Formulae by GILLES DOWEK AND YING JIANG |
| We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F. |
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17.10.38053 Mateusz Pabian |
Algorytmy Randomizowane i Aproksymacyjne New approximation algorithm for (1,2)-TSP (Adamaszek, Mnich, Paluch) |
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05.04.24364 Marcin Muszalski |
Optymalizacja Kombinatoryczna On the queue-number of graphs with bounded tree-width |
In this talk I will present upper bound for a queue-number of graphs with bounded tree-width obtained by Veit Wiechert. The new upper bound, 2k - 1, improves upon double exponential upper bounds due to Dujmović et al. and Giacomo et al. Additionally I will show his construction of k-trees that have queue-number at least k + 1. The construction solves a problem of Rengarajan and Veni Madhavan, namely, that the maximal queue-number of 2-trees is equal to 3. Marcin Muszalski. Queue-number of graphs with bounded tree-width - Veit Wiechert. slides. 2018. |
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25.05.24305 Weronika Grzybowska, Paweł Mader |
Hamming distance completeness and sparse matrix multiplication |
| Autorzy prezentują polilogarytmiczne redukcje pomiędzy obliczaniem odległości Hamminga a iloczynem skalarnym, w którym miejsce mnożenia zajmuje pewna funkcja binarna na liczbach całkowitych. Dla takich funkcji binarnych należą dominance product, threshold product i odległości l_{2p+1} dla stałego p. Wykorzystując wyżej opisane redukcje, autorzy wykazują równość (z dokładnością do czynników polilogarytmicznych) złożoności wyliczania powyższych funkcji dla dwóch zbiorów wektorów. Dodatkowo, autorzy dowodzą, że APHam (oraz ten sam problem z użyciem innych wymienionych funkcji) mieści się w czasie polilogarytmicznym od mnożenia macierzy rozmiaru n na nd i nd na n, zawierających po nd niezerowych wartości. |
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25.07.21516 Paweł Palenica |
Podstawy Informatyki On Randomised Strategies in the λ-Calculus by Ugo Dal Lago and Gabriele Vanoni |
| In this work we introduce randomized reduction strategies - a notion already studied in the context of abstract reduction systems - for the lambda-calculus. We develop a simple framework that allows us to prove if a probabilistic strategy is positive almost-surely normalizing. Then we propose a simple example of probabilistic strategy for the lambda-calculus that has such a property and we show why it is non-trivial with respect to classical deterministic strategies such as leftmost-outermost or rightmostinnermost. We conclude studying this strategy for two classical sub- lambda calculi, namely those duplication and erasure are syntactically forbidden. |
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11.06.18888 Szymon Łukasz |
Algorytmy Randomizowane i Aproksymacyjne NP-hardness of coloring 2-colorable hypergraph with poly-logarithmically many colors (A. Bhangale) |
We give very short and simple proofs of the following statements: Given a 2-colorable 4-uniform hypergraph on n vertices, 1) It is NP-hard to color it with log^delta n colors for some delta>0. 2) It is quasi-NP-hard to color it with O({log^{1-o(1)} n}) colors. |
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31.03.87226 Rafał Burczyński |
Podstawy Informatyki A Hitchhiker’s Guide to descriptional complexity through analytic combinatorics by Sabine Broda, António Machiavelo, Nelma Moreira and Rogério Reis |
| Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in this area without too much effort. In this gentle introduction we take the reader on a journey through the basic analytical tools of that theory, giving some illustrative examples using regular expressions. Additionally, new asymptotic average-case results for several $\epsilon-NFA$ constructions are presented, in a unified framework. It turns out that, asymptotically, and in the average case, the complexity gap between the several constructions is significantly larger than in the worst case. Furthermore, one of the $\epsilon-NFA$ constructions approaches the corresponding $\epsilon-free NFA$ construction, asymptotically and on average. |
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16.02.84598 Wiktor Daniec |
Algorytmy Randomizowane i Aproksymacyjne David Galvin, “Three tutorial lectures on entropy and counting” (rozdział 5) |
| David Galvin, “Three tutorial lectures on entropy and counting” (rozdział 5) |
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05.08.70908 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna A new variant of the game of cops and robber |
We consider the following metric version of the Cops and Robbers game. Let G be a simple graph and let k≥1 be a fixed integer. In the first round, Cop picks a subset of k vertices B={v1,v2,...,vk} and then Robber picks a vertex u but keeps it in a secret. Then Cop asks Robber for a vector Du(B)=(d1,2,...,dk) whose components di=dG(u,vi), i=1,2,...,k, are the distances from u to the vertices of B. In the second round, Robber may stay at the vertex u or move to any neighbouring vertex which is kept in a secret. Then Cop picks another k vertices and asks as before for the corresponding distances to the vertex occupied by Robber. And so on in every next round. The game stops when Cop determines exactly the current position of Robber. In that case, she is the winner. Otherwise, Robber is the winner (that is if Cop is not able to localise him in any finite number of rounds). Let ζ(G) denote the least integer k for which Cop has a winning strategy. Notice that this parameter is well defined since the inequality ζ(G)≤|V(G)| holds obviously. The aim of the talk is to present results concerning 2-trees, outerplanar graphs and planar graphs. This is a joint work with Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, and Małgorzata Śleszyńska-Nowak. |
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23.09.70849 Filip Bartodziej, Vladyslav Hlembotskyi |
Fine-grained Lower Bounds on Cops and Robbers |
| Sumienni policjanci, czy sprytny złodziej? Na tym seminarium dowiemy się kto triumfuje, jak szybko (lub jak wolno) jesteśmy w stanie się o tym przekonać i ilu policjantów wystarczy, aby przyskrzynić nawet samego Frank’a Abagnale’a. Rozważania bedą oparte o grę strategiczna w policjantów i złodziei na grafie (cops and robbers). Uzyskane wyniki opierają sie na założeniu SETH/ETH. |
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23.11.68060 Szymon Stankiewicz |
Podstawy Informatyki Encoding Turing Machines into the Deterministic Lambda Calculus by Ugo Dal Lago and Beniamino Accattoli |
This note is about encoding Turing machines into the lambda -calculus. The encoding we show is interesting for two reasons: 1. Weakly strategy independent : the image of the encoding is a very small fragment of the lambda - calculus, that we call the deterministic lambda -calculus det. Essentially, it is the CPS (continuation-passing style) lambda -calculus restricted to weak evaluation (i.e., not under abstractions). In det every term has at most one redex, and so all weak strategies collapse into a single deterministic evaluation strategy, because there are no choices between redexes to be made. The important consequence of this property is that every weak evaluation strategy then allows to simulate Turing machines,as well as any strong strategy reducing weak head redexes (or even only weak head redexes) first. 2. Linear overhead: the simulation is very efficient, when taking the number of beta-steps as the time cost model for the deterministic lambda -calculus. The simulation in det indeed requires a number of beta-steps that is linear in the number of transitions of the encoded Turing machine, which is the best possible overhead. Therefore, not only all weak strategies simulate Turing |
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31.03.51743 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Local Dimension is Unbounded for Planar Posets |
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetril and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height. This is a joint work with Jarosław Grytczuk and W.T. Trotter. |
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18.05.51684 Jan Mełech, Rafał Burczyński |
A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum |
| Celem znanego problemu NP-zupełnego Subset-Sum jest znalezienie takiego podzbioru multizbioru o mocy n, którego suma elementów wynosi t. Autorzy prezentują krótkie probabilistyczne rozwiązanie bazujące na szybkiej transformacie Fouriera oraz manipulacjach na funkcjach tworzących działające w czasie O((n+t)*polylog(t)) i zwracające odpowiedź z prawdopodobieństwem błędu rzędu O(1/(n+t)). Ten wynik został osiągnięty wcześniej, jednak praca upraszcza rozwiązanie, zawierając je raptem w kilku stronach. |
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04.05.49005 08.09.68170 Patryk Mikos |
Informatyka Teoretyczna Does the representation matter? |
The class of unit interval graphs has at least 3 equivalent definitions:
We ask whether the competitive ratio in the online unit-interval graph coloring with bandwidths depends on the chosen graph representation. |
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19.07.48895 Vladyslav Hlembotskyi |
Podstawy Informatyki Limited Automata and Regular Languages by Giovanni Pighizzini and Andrea Pisoni |
Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d \geq 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schutzenberger representation for context-free languages,
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23.11.32577 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna A Tight Bound for Shortest Augmenting Paths on Trees |
The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T=(WB, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et. al. that the total length of all shortest augmenting paths found is O(n log n). In this paper we prove a tight O(n log n) upper bound for the total length of shortest augmenting paths for trees improving over O(n log² n) bound. This is a joint work with Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz. |
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12.01.32519 Dawid Pyczek, Michał Zieliński |
On the Worst-Case Complexity of TimSort |
| TimSort jest bardzo interesującym algorytmem sortującym, który został wprowadzony do Pythona stosunkowo niedawno, bo w 2002 roku. Ten bardzo popularny algorytm używany jest z powodzeniem na całym świecie. Wynika to z faktu, że działa on wyjątkowo szybko na częściowo posortowanych danych. Aż do niniejszej pracy nie była znana pesymistyczna złożoność tego algorytmu -- w pracy pokazane zostanie, że pesymistyczna złożoność algorytmu TimSort wynosi O(n log n). Następnie złożoność algorytmu ograniczymy przez O(n+n log ρ), gdzie ρ to ilość przebiegów. Pierwsza złożoność w bezpośredni sposob wynika z drugiej, ale oba dowody są ciekawe i pomagają lepiej zrozumieć działanie TimSorta. Dodatkowo w wyniku analizy algorytmu autorzy pracy odryli błąd w implementacji TimSorta w Javie. |
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28.12.29839 Andrzej Dorobisz |
Informatyka Teoretyczna Induced subgraphs of graphs with large chromatic number |
Based on the paper a proof of a 1985 conjecture of Gyarfas that for all k, ℓ, every graph with sufficiently large chromatic number contains either a clique of cardinality more than k or an induced cycle of length more than ℓ will be presented. |
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14.03.29730 Michał Zieliński |
Podstawy Informatyki Lambda Theories allowing Terms with a Finite Number of Fixed Points by BENEDETTO INTRIGILA and RICHARD STATMAN |
| A natural question in the lambda calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in lambda-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are lambda-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) lambda-theory where the range property is violated. A second, more complex example of a non-r.e. Lambda-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term. |
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06.11.10564 Jarosław Duda Instytut Informatyki UJ |
Podstawy Informatyki Krótkie wprowadzenie do ANS, MERW i pól Markova |
| Na seminarium spróbuję zainteresować kilkoma z tematów, którymi się zajmowałem, np. kodowaniem Asymmetric Numeral Systems, które jest obecnie używane w produktach m.in. Apple, Facebook, Google. Opowiem też o Maximal Entropy Random Walk, czyli jak optymalnie wybierać błądzenie przypadkowe na grafie - z perspektywy zastosowań m.in. do maksymalizacji ilości przechowywanej informacji, zrozumienia i naprawienia rozbieżności między dyfuzją a mechaniką kwantową, analizy obrazów, sieci społecznych, czy rekonstrukcji traktów nerwowych. Tematem łączącym powyższe będą pola Markova, czyli wielowymiarowe uogólnienie procesów Markova, o których też krótko opowiem z przykładem zastosowania do poprawienia pojemności dysków twardych. Slajdy do seminarium można znaleźć na: http://tiny.cc/2jpiyy |
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21.06.2018 |
Wykład Wojciecha Szpankowskiego "Analytic Information Theory: From Shannon to Knuth and Back" |
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14.04.40791 Mateusz Twaróg, Patryk Urbański, Łukasz Majcher |
Optymalizacja Kombinatoryczna Progress in the Arachne Project |
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01.10.37943 Marcin Briański |
Podstawy Informatyki COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS by DENIS HIRSCHFELDT, CARL JOCKUSCH, RUTGER KUYPER, AND PAUL SCHUPP |
A coarse description of a set A \subset \omega is a set D \subset \omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1- genericity in place of weak 2-randomness. In the other direction, we show that if A \leq_T \emptyset is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set.
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24.04.21653 Krzysztof Maziarz |
Optymalizacja Kombinatoryczna The chromatic number of the plane is at least 5 |
The Hadwiger-Nelson problem asks for the minimum number of colors required to color the plane, in such a way, that any two points at distance exactly one are assigned different colors. Albeit its simple definition, no significant progress on the question was made for nearly a century. In the discussed paper, Aubrey D. N. J. de Grey has shown a set of points in the plane, such that 5 colors are necessary to color it properly, thus improving a long-standing lower bound of 4 colors. Interestingly, the smallest such set discovered so far has 1581 vertices. The chromatic number of the plane is at least 5, Aubrey D.N.J. de Grey |
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07.12.21625 Szymon Łukasz |
Optymalizacja Kombinatoryczna Dynamic F-free Coloring of Graphs |
An F-free coloring is a coloring of a graph such that each color induces an F-free graph. In this talk we consider dynamic F-free coloring which can be interpreted as a game of Presenter and Painter. In each move Presenter presents new vertices along with the edges between them and already known vertices. In the same move Presenter can also discolor arbitrary vertices and request Painter to color some vertices. The problem we consider can be stated as follows: For a given graph G, is there a sequence of moves for which the greedy algorithm uses at least k colors during dynamic F-free coloring of G. We will show that for some classes of graphs this problem is decidable in polynomial time (for fixed F and k) in the case where F is 2-connected or F is path of length 2. Piotr Borowiecki, Elżbieta Sidorowicz, Dynamic F-free Coloring of Graphs, Graphs and Combinatorics 2018, Volume 34, Issue 3, pp 457-475 |
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26.05.18778 Bruno Pitrus |
Podstawy Informatyki Linear lambda terms as invariants of rooted trivalent maps by Noam Zeilberger |
The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between \alpha equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.
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16.09.84597 10.01.18888 Grzegorz Herman |
Informatyka Teoretyczna Relational parsing: a clean generalized parsing algorithm. |
We propose a new, worst-case cubic-time, generalized parsing algorithm for all context-free languages, based on computing the relations between configurations and transitions in a recursive transition network. The algorithm represents such relations using abstract data types, and for their specific (non-canonical) implementations behaves analogously to generalized LL, Left-Corner, or LR. It features a clean mathematical formulation, and can easily be implemented using only immutable data structures. |
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30.01.84488 Bartłomiej Puget |
Podstawy Informatyki STATMAN'S HIERARCHY THEOREM by BRAM WESTERBAAN, BAS WESTERBAAN, RUTGER KUYPER, CARST TANKINK, REMY VIEHOFF AND HENK BARENDREGT |
In the Simply Typed lambda calculus Statman investigates the reducibility relation between types: for types freely generated using \arrow and a single ground type 0, define A \leq B if there exists a lambda definable injection from the closed terms of type A into those of type B. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) \omega + 4.
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08.04.68170 Marcin Briański |
Optymalizacja Kombinatoryczna How many ants does it take to find the food? |
In this talk we will consider the ANTS (Ants Nearby Treasure Search) problem: consider n agents (ants), controlled by finite automata (or PDAs) exploring an infinite grid attempting to locate a hidden treasure. The question we want to answer is: how many agents are necessary to accomplish this task in (expected) finite time? Of course, the answer will depend on the way we model this situation. We will consider synchronous as well as asynchronous environment, agents with access to randomness as well as deterministic ones, agents controlled by PDA as well as finite automata and various combinations thereof. In most cases established bounds are tight, however in certain cases there is still ample room for improvement (which some might consider interesting). Yuval Emek, Tobias Langner, David Stolz, Jara Uitto, Roger Wattenhofer, How many ants does it take to find the food?, Theoretical Computer Science Volume 608, Part 3, 10 December 2015, Pages 255-267 |
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02.12.49004 Marcin Muszalski |
Optymalizacja Kombinatoryczna On the Number of Maximum Empty Boxes Amidst n Points |
I will present article written by Adrian Dumitrescu and Minghui Jiang in which they revisit the following problem (along with its higher dimensional variant): |
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22.03.46157 Maciej Czerwiński |
Podstawy Informatyki On Type Inference in the Intersection Type Discipline by Gerard Boudol and Pascal Zimmer |
| We introduce a new unification procedure for the type inference problem in the intersection type discipline. We show that unification exactly corresponds to reduction in an extended lambda calculus, where one never erases arguments that would be discarded by ordinary β-reduction. We show that our notion of unification allows us to compute a principal typing for any strongly normalizing lambda expression. |
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28.07.29839 Jakub Szarawski |
Optymalizacja Kombinatoryczna Faster approximation schemes for the two-dimensional knapsack problem |
In 2008 Klaus Jansen and Roberto Solis-Oba presented a polynomial time approximation scheme (PTAS) for the square packing problem. Sandy Heydrich and Andreas Wiese base on their work and show a faster approximation (EPTAS) for the same problem. During the seminar both the common parts of the two papers (such as dividing the squares into large and small ones, dividing the rectangle into cells, frames, rows and blocks) and the new ideas (faster large squares guessing and block size guessing) will be presented. |
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15.11.26991 Dominika Salawa |
Podstawy Informatyki The Hiring Problem and Permutations by Margaret Archibald and Conrado Martínez |
The hiring problem has been recently introduced by Broder et al. in last year’s ACM-SIAM Symp. on Discrete Algorithms (SODA 2008), as a simple model for decision making under uncertainty. Candidates are interviewed in a sequential fashion, each one endowed with a quality score, and decisions to hire or discard them must be taken on the fly. The goal is to maintain a good rate of hiring while improving the “average” quality of the hired staff. We provide here an alternative formulation of the hiring problem in combinatorial terms. This combinatorial model allows us the systematic use of techniques from combinatorial analysis, e. g., generating functions, to study the problem. |
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31.12.73645 Sylwester Klocek |
Optymalizacja Kombinatoryczna Colouring of (P3∪P2)-free graphs |
In a paper authors are colouring of (P3∪P2)-free graphs, a super class of 2K2 -free graphs. During lecture I am going to present three discovered upper bounds of the chromatic number of (P3∪P2) -free graphs, and sharper bounds for (P3∪P2 , diamond)-free graphs and for (2K2, diamond)-free graphs. The first part of a talk will contain an explanation of terminology and notation along with problem statements and results. In the second part, I will focus on proving each result in a sequence of claims and proofs. Arpitha P. Bharathi, Sheshayya A. Choudum, Colouring of (P3∪P2)-free graphs, Graphs and Combinatorics, Volume 34 (1), 2018 |
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04.02.70908 Jacek Krzaczkowski |
Informatyka Teoretyczna Complexity of solving equations |
Solving equations is one of the oldest and well known mathematical problems which for centuries was the driving force of research in algebra. Let us only mention Galois theory, Gaussian elimination or Diophantine Equations. If we consider equations over the ring of integers it is the famous 10th Hilbert Problem on Diophantine Equations, which has been shown to be undecidable. In finite realms such a problem is obviously decidable in nondeterministic polynomial time. The talk is intended to present the latest achievements in searching structural algebraic conditions a finite algebra A has to satisfy in order to have a polynomial time algorithm that decides if an equation of polynomials over A has a solution. We will also present the results on the polynomial equivalence problem in which we ask whether two polynomials over a finite algebra describe the same function. This is joint work with Paweł M. Idziak and Piotr Kawałek.. |
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21.06.70794 Rafał Burczyński |
Podstawy Informatyki How to select a loser |
| Consider the following game: everyone from a group of n people flips a coin with result either 0 or 1, both equally probable; if no one gets a 0, the round is repeated, otherwise people with 1's are considered "winners" and the game continues only with participants who got 0's. The process continues until there is one person left, who is called "loser". We can picture this process as a binary tree and analyze some of its properties in average case. The analysis is not completely trivial, in particular one may find application for tools such as Mellin transform. |
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25.08.54480 Grzegorz Jurdziński |
Optymalizacja Kombinatoryczna Split Packing: An Algorithm for Packing Circles with Optimal Worst-Case Density |
Circle packing problem, where one asks whether a given set of circles can be fit into a unit square, is known to be NP-hard. I will show that when combined area of circles does not exceed ≈0,539, then it is possible to pack them. The given bound is tight in the meaning that for larger combined area an instance impossible to pack can be found. Proof for this theorem is constructive and gives an algorithm, called Split Packing, for finding a solution for instances satisfying the conditions. Moreover it can also serve as a constant-factor approximation algorithm for the problem of finding a smallest square which can fit given circles. |
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13.02.51629 Rafał Burczyński |
Podstawy Informatyki Mellin transforms and asymptotics |
| We will introduce Mellin transform, which may be used for the asymptotic analysis of a particular class of sums. I will start with basic properties and then present fundamental correspondence between the asymptotic expansion of a function at 0 or infinity and singularities of its transform. Finally we will show some combinatorial applications of the transform. |
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21.04.35315 Maciej Woźniak |
Optymalizacja Kombinatoryczna Find Your Place: Simple Distributed Algorithms for Community Detection |
Graph G = (V_1 \cup V_2, E) is regular clustered graph (with two communities) if:
We define (weak) block reconstruction of graph as two-coloring of vertices that separates V_1 and V_2 up to small "error" fraction of vertices. The reconstruction is said to be strong if separation is exact. I will present simple distributed algorithm (protocol) that produces strong reconstruction for clustered regular graphs within O(log n) iterations. I will also show that this algorithm produces weak reconstruction for non-regular clustered graphs with two communities and discuss an approach to solving this problem for regular graphs with more than two communities. |
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09.08.32467 Weronika Grzybowska |
Podstawy Informatyki Average complexity of Moore’s and Hopcroft’s algorithms by Julien David |
| In this paper we prove that for the uniform distribution on complete deterministic automata, the average time complexity of Moore’s state minimization algorithm is O(n log (log n)), where n is the number of states in the input automata and the number of letters in the alphabet is fixed. Then, an unusual family of implementations of Hopcroft’s algorithm is characterized, for which the algorithm will be proved to be always faster than Moore’s algorithm. Finally, we present experimental results on the usual implementations of Hopcroft’s algorithm. |
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14.12.16149 Anna Kobak |
Optymalizacja Kombinatoryczna On tree-partition-width |
A tree-partition of a graph G is a proper partition of its vertex set into "bags", such that identifying the vertices in each bag produces a forest. The width of a tree-partition is the maximum number of vertices in a bag. The tree-partition-width of G is the minimum width of a tree-partition of G. I will prove three theorems presented in the article, showing an upper bound on the tree-partition-width of all graphs, a lower bound for chordal graphs and a lower bound for graphs with tree-width 2. |
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18.01.13412 Bartosz Walczak |
Informatyka Teoretyczna Sparse Kneser graphs are Hamiltonian |
For integers Joint work with Torsten Mütze and Jerri Nummenpalo (arXiv:1711.01636). |
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04.04.13302 Vladyslav Hlembotskyi |
Podstawy Informatyki A graph theoretic approach to automata minimality by Antonio Restivo and Roberto Vaglica |
| The paper presents a graph-theoretic approach to test the minimality of a deterministic automaton. In particular, we focus on problems concerning the dependence of the minimality of an automaton on the choice of the set F of final states or on the cardinality of the set F . We introduce different minimality conditions of an automaton and show that such conditions can be characterized in graph-theoretic terms. |
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24.09.79121 Grzegorz Guśpiel |
Informatyka Teoretyczna On the Complexity of Crossing Minimization |
For a bipartite graph G with vertex bipartition (X, Y), a two-layer drawing of G (on the plane) is a placement of vertices in X and Y in distinct points on two parallel lines LX and LY, respectively. Then, each edge is drawn by connecting its end vertices by a straight line segment. A bipartite graph with a two-layer drawing is a two-layered graph. We study basic graph problems on two-layered graphs, where the goal is to minimize the number of pairwise crossing edges in the graph structure we seek. The graph structure can be a perfect matching, a Hamiltonian path or an (s, t)-path. We investigate the complexity of these problems, obtaining some hardness proofs, FPT algorithms and small kernels.
This is joint work with Akanksha Agrawal, Jayakrishnan Madathil, Saket Saurabh and Meirav Zehavi. |
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09.02.79008 Szymon Stankiewicz |
Podstawy Informatyki Introduction to Higher-Order Categorical Logic - continuation |
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15.04.62694 Aleksandra Mędrek |
Optymalizacja Kombinatoryczna The Matching Problem in General Graphs is in Quasi-NC |
Authors show that the perfect matching problem in general graphs is in quasi-NC by presenting a deterministic parallel algorithm which runs in O(log^3 n) time on n^O(log^2 n) processors. The paper extends the framework of Fenner, Gurjar and Thierauf, who proved that finding perfect matching in bipartite graphs is in quasi-NC. I describe their algorithm in the first part of my presentation. In the second part I talk about difficulties that arise in the general case and how they are approached. Ola Svensson, Jakub Tarnawski, The Matching Problem in General Graphs is in Quasi-NC, FOCS 2017 |
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04.10.59842 Szymon Stankiewicz |
Podstawy Informatyki Introduction to Higher-Order Categorical Logic by Lambec and Scott |
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09.12.43528 Dawid Pyczek |
Optymalizacja Kombinatoryczna Punctured combinatorial Nullstellensätze |
This article presents an extension of Alon’s Nullstellensatz to functions of multiple zeros at the common zeros of some polynomials. It also includes an introduction to the polynomials of multiple variables and other useful definitions. There are also many corollaries useful for polynomial problem-solving. Possibly the presentation will include some geometrical usage of Nullstellensatze extensions. |
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12.01.40791 Michael Kompatscher Charles University in Prague |
Informatyka Teoretyczna CSPs of infinite structures and equations in oligomorphic algebras |
In 1998 Feder and Vardi famously conjectures that the constraint satisfaction problem (CSP) of a finite structure is either in P or NP-complete. Universal algebra turned out to be the main tool in tackling their conjecture and lead to two recent proofs, showing that CSP(A) is in P if the polymorphism algebra associated with A is a Taylor algebra, and NP-complete otherwise.
For CSPs of structures with infinite domains this universal algebraic approach fails badly in general. However, if the automorphism group of the structure is "sufficiently big", i.e. oligomorphic, many results can be transferred from the finite case. We survey results about the equational structure of oligomorphic algebras and their applications to constraint satisfaction problems. |
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27.05.40681 Dawid Pyczek i Jakub Rówiński |
Podstawy Informatyki Asymptotic Density and the Theory of Computability by CARL JOCKUSCH AND PAUL SCHUPP |
| The purpose of this paper is to survey recent work on how classical asymptotic density interacts with the theory of computability. We have tried to make the survey accessible to those who are not specialists in computability theory and we mainly state results without proof, but we include a few easy proofs to illustrate the flavor of the subject. In complexity theory, classes such as P and NP are defined by using worst-case measures. That is, a problem belongs to the class if there is an algorithm solving it which has a suitable bound on its running time over all instances of the problem. Similarly, in computability theory, a problem is classified as computable if there is a single algorithm which solves all instances of the given problem. There is now a general awareness that worst-case measures may not give a good picture of a particular algorithm or problem since hard instances may be very sparse. The paradigm case is Dantzig’s Simplex Algorithm for linear programming problems. This algorithm runs many hundreds of times every day for scheduling and transportation problems, almost always very quickly. There are clever examples of Klee and Minty showing that there exist instances for which the Simplex Algorithm must take exponential time, but such examples are not encountered in practice. Observations of this type led to the development of average-case complexity by Gurevich and by Levin independently. There are different approaches to the average-case complexity, but they all involve computing the expected value of the running time of an algorithm with respect to some measure on the set of inputs. Thus the problem must be decidable and one still needs to know the worst-case complexity. Another example of hard instances being sparse is the behavior of algorithms for decision problems in group theory used in computer algebra packages. There is often some kind of an easy “fast check” algorithm which quickly produces a solution for “most” inputs of the problem. This is true even if the worst-case complexity of the particular problem is very high or the problem is even unsolvable. Thus many group-theoretic decision problems have a very large set of inputs where the (usually negative) answer can be obtained easily and quickly. |
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30.12.5511097 Wojciech Szpankowski Purdue University USA |
Podstawy Informatyki Analytic Information Theory: From Shannon to Knuth and Back |
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04.08.24363 Jakub Rówiński |
Optymalizacja Kombinatoryczna On the 1/3–2/3 Conjecture |
Let (P,≤) be a finite poset. For distinct elements x, y ∈ P , we define P(x ≺ y) to be the proportion of linear extensions of P in which x comes before y. For 0 ≤ α ≤ 1, we say (x,y) is an α-balanced pair 2 if α ≤ P(x ≺ y) ≤ 1 − α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. Proof of above conjecture as well as stronger condition of having a 1/2-balanced pair for certain families of posets will be shown. These include lattices such as the Boolean, set partition, subspace lattices and variety of diagrams. Emily J. Olson, Bruce E. Sagan, On the 1/3--2/3 Conjecture, Order, 2018 |
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21.01.21516 Jarosław Duda Instytut Informatyki UJ |
Podstawy Informatyki Some nonstandard approaches to hard computational problems |
I will talk about nonstandard approaches to some problems for which there is not known polynomial time classical algorithm. I will start with briefly explaining mechanism used in Shor algorithm, compressed sensing, and the problem with global optimization formulations used in adiabatic
Slides: https://tinyurl.com/y74nx2t6 |
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27.08.79121 Piotr Micek |
Informatyka Teoretyczna Seymour's conjecture on 2-connected graphs of large pathwidth |
We prove the conjecture of Seymour (1993) that for every apex-forest H1 and outerplanar graph H2 there is an integer p such that every 2-connected graph of pathwidth at least p contains H1 or H2 as a minor. This is joint work with Tony Huynh, Gwenaël Joret, and David R.Wood. |
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06.11.70907 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna News about Combinatorial Nullstellensatz |
I will present some new theorems, proofs and open problems concerning about Combinatorial Nullstellensatz and related problems. |
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01.07.51742 Jarek Duda |
Optymalizacja Kombinatoryczna Some nonstandard approaches to hard computational problems |
I will talk about nonstandard approaches to some problems for which there is not known polynomial time classical algorithm. I will start with briefly explaining mechanism used in Shor algorithm and the problem with global optimization formulations used in adiabatic quantum computers. Then show some perspectives on the subset-sum NP complete problem, like geometric, integration and divergence formulations. Then show how Grassmann variables would be useful for the Hamilton cycle problem. Finally discuss the difficulty of the graph isomorphism problem on the most problematic cases: strongly regular graphs, and algebraic perspective on this problem. Jarek Duda. Some unusualapproaches to hard computational problems. slides. |
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04.08.49004 09.12.68169 Andrzej Dorobisz |
Informatyka Teoretyczna Online bipartite matching with amortized O(log²n) replacements |
In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log²n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω(log n) lower bound. The previous best known strategy achieved amortized O(√n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014].
Based on the paper: Online bipartite matching with amortized O(log²n) replacements by Aaron Bernstein, Jacob Holm and Eva Rotenberg |
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19.12.48890 Bartłomiej Puget i Dominika Salawa |
Podstawy Informatyki Chapters 8.5 - 8.9 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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23.02.32577 Maciej Woźniak, Dawid Pyczek |
Optymalizacja Kombinatoryczna Online Vertex Cover and Matching: Beating the Greedy Algorithm |
Authors study the online vertex cover problem and online matching problem in bipartite graphs and in general graphs. For the case of bipartite graphs their result is optimal water-filling algorithm with competitive ratio 1/(1-1/e) . The main contribution of this paper is a 1.901-competitive algorithm for vertex cover in general graphs which beats the well-known trivial 2-competitive algorithm. The next major result is a primal-dual analysis of given algorithm that implies the dual result of a 0.526-competitive algorithm for online fractional matching in general graphs. On the hardness side authors show that no randomized online algorithm can achieve a competitive ratio better than 1.753 and 0.625 for the online fractional vertex cover problem and the online fractional matching problem respectively, even for bipartite graphs. |
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14.08.29725 Kamil Rajtar |
Podstawy Informatyki Chapters 8.1 - 8.4 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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20.10.13411 Grzegorz Bukowiec |
Optymalizacja Kombinatoryczna Feedback Vertex Set Problem |
A Feedback Vertex Set (FVS) is a subset of vertices in a graph such that its removal results in an acyclic graph. The problem of finding a minimal FVS is one of the classic NP-complete problems. However, in some practical cases, we can assume that its size is fairly small. This motivated an intensive study of the parametrized version of this problem, which asks either for FVS of a size at most k or an information that it doesn't exist. There are several deterministic algorithms known which solve this in time O*(ck), the best one for now being O*(3.592k). |
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08.04.10560 Dawid Pyczek i Jakub Rowiński |
Podstawy Informatyki Chapters 7.6 - 7.9 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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19.02.59956 Paweł Kubiak, Jakub Rówiński |
Optymalizacja Kombinatoryczna Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms |
On bipartite graphs, problem of constrained minimum vertex cover (MIN-CVCB) is defined as follows: given a bipartite graph G = (V, E) with vertex bipartition V = U ∪ L and two integers ku and kl, decide whether there is a minimum vertex cover in G with at most ku vertices in U and at most kl vertices in L. We show how it is related to practical problems. We prove that (MIN-CVCB) is NP-complete. However, there are many parametrized algorithms running in decent time. We describe one of them, whereby linear kernelization method it achieves O(1.26ku+kl +(ku +kl)|G|) time. |
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24.03.57218 Grzegorz Herman |
Informatyka Teoretyczna Declarative name resolution for complex, extensible languages |
We present a new, declarative, language-independent model for name resolution, based on a data flow graph built using simple combinators. The model is expressive enough to capture complex name binding rules of modern programming languages (e.g., partial definitions, C++ argument-dependent lookup). It is also designed to make it straightforward toextend a language with new syntactic constructs, including new categories of names. The model, together with a proof-of-concept resolution engine, has been implemented in Haskell, and evaluated on syntax trees of C# programs.
(This is joint work with Katarzyna Bułat.)
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09.08.57104 Rafał Burczyński |
Podstawy Informatyki Chapters 7.1 - 7.5 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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21.01.40814 Jakub Nowak |
Optymalizacja Kombinatoryczna Dulmage–Mendelsohn Decomposition |
In a graph G, let B be the set of vertices covered by every maximum matching in G, and let D = V(G) − B. Further partition B by letting A be the subset consisting of vertices with at least one neighbor outside B, and let C = B − A. The Gallai-Edmonds Decomposition of G is the partition of V(G) into the three sets A, C, D. The Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and only if they are paired with each other in a perfect matching of the graph. It is an extension of the Gallai-Edmonds decomposition. L. Lovász, M. D. Plummer. Matching theory. North-Holland Mathematics Studies, 121. Annals of Discrete Mathematics, 29. North-Holland Publishing Co., Amsterdam. 1986. pp. xxvii+544. ISBN: 0-444-87916-1. Chapter 4.3. |
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14.10.40790 Lev Deliatynskyi |
Optymalizacja Kombinatoryczna A short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem |
This paper studies the maximum matching in a graph. It shows a short proof of a Berge-Tutte formula and the Gallai-Endmonds structure theorem. Authors use Hall's theorem to prove it. Deficiency in a graph (def(S), S⊆V(G)) is o(G-S) - |S|, where o(G-S) is the number of odd components in G-S. Berge-Tutte formula says that the maximum size of a matching in an n-vertex graph G is 1/2(n-def(G)), where def(G) = maxS⊆V(G)def(S). Gallai Edmonds has a sharper formulation which gives considerable information about the structure of maximum size matchings. |
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16.11.38052 Tony Huynh Universite de Libre Bruxelles |
Informatyka Teoretyczna Strengthening Convex Relaxations of 0/1-Sets using Boolean Formulas |
In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. |
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04.04.37939 Katarzyna Grzybowska |
Podstawy Informatyki Chapters 6.12 - 6.15 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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08.06.21625 Jan Derbisz, Franciszek Stokowacki |
Optymalizacja Kombinatoryczna On Low Rank-Width Colorings |
We say that a class C of graphs admits low rank-width colorings if there exist functions N : N → N and Q: N → N such that for all p ∈ N, every graph G ∈ C can be vertex colored with at most N(p) colors such that the union of any i ≤ p color classes induces a subgraph of rank-width at most Q(i). It turns out that for every graph class C of bounded expansion and every positive integer r, the class {Gr : G ∈ C} of r-th powers of graphs from C, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. Additionally, every graph class admitting low rank-width colorings is χ-bounded. |
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27.11.18773 Katarzyna Bułat |
Podstawy Informatyki Chapter 6.8 - 6.11 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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19.03.84597 Krzysztof Maziarz, Tomasz Wesołowski |
Optymalizacja Kombinatoryczna The Generalised Colouring Numbers on Classes of Bounded Expansion |
We introduce two classes of graphs - graphs with bounded expansion and nowhere dense graphs. These notions are a common generalization of proper minor closed classes, classes of graphs with bounded degree, locally planar graphs, to name just a few classes which are studied extensively in combinatorial and computer science contexts. We also present generalized colouring numbers admr(G), colr(G), and wcolr(G) and show important applications in the theory of above-mentioned classes of graphs. Finally, we prove that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of r, and show that it can be efficiently computed. |
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23.04.81859 Adam Polak |
Informatyka Teoretyczna Open problems in algorithms and complexity |
During the talk I'll present several interesting open problems, including, but not limited to:
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07.09.81745 Filip Bartodziej |
Podstawy Informatyki Chapter 6.1 - 6.7 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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13.11.65431 Gabriel Jakóbczak |
Optymalizacja Kombinatoryczna Majority coloring games |
A vertex coloring of graph G satisfies the majority rule, if for each vertex v at most half of its neighbors receive the same color as v. A coloring which satisfies the majority rule is called majority coloring. We consider its game version. For given graph G and color set C two players, Alice and Bob, in alternating turns color vertices with respect to the majority rule. Alice wins when every vertex becomes colored, while goal for Bob is to create a vertex which cannot be colored with any color belonging to the set C without breaking the majority rule. Let µg(G) denote the least number of colors belonging to C for which Alice has winning strategy in game on graph G. We show that if the color set C is finite, there exists a graph G on which Bob has winning strategy. We prove also that for graphs with col(G) = 3 parameter µg(G) is still unbounded. |
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16.12.62693 Patryk Mikos |
Informatyka Teoretyczna On-line interval coloring for bounded length intervals |
On-line interval coloring was studied by Kierstead and Trotter. They presented an algorithm with competitive ratio 3,and showed a construction which implies that there is no algorithm with competitive ratio strictly less than 3. However, their construction in asymptotic case requires intervals with possibly unbounded length. We are interested in a variant of the on-line interval coloring problem in which all intervals have lenght between 1 and L. We show that as L tends to infinity the asymptotic competitive ratio tends to 5/2. Moreover we show that for L=1+epsi there is no algorithm with competitive ratio less than 5/3 and for L=2+epsi there is no algotihm with competitive ratio less than 7/4. Finally, we want to know how the asymptotic competitive ratio changes as a function of L. |
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02.05.62580 Michał Ziobro |
Podstawy Informatyki Chapter 5 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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08.07.46266 Anna Kobak, Grzegorz Jurdziński |
Optymalizacja Kombinatoryczna The Erdős discrepancy problem - Part II |
Erdős discrepancy problem has waited for the solution for over 70 years until last year Terrence Tao, with a help of Polymath project, has published a paper with its solution. After having our friends given an introduction to the topic and shown the Fourier analytic reduction of the problem last week we will continue presenting the proof. It will include the proof of Elliot-type conjecture and a sketch of how to apply a generalised Borwein-Choi-Coons analysis for the final steps of the main proof. Terence Tao. The Erdős discrepancy problem. Discrete Analysis. Vol. 2 (2016), pp. 1-20. |
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11.08.43528 Tomasz Krawczyk |
Informatyka Teoretyczna Representation and coloring of partially ordered sets under conditions of incomplete information |
The purpose of my talk is to discuss several problems related to coloring and construction of appropriate representation for partially ordered sets (also posets) and graph classes derived from posets. In these problems, we will assume the following two scenarios: in the first scenario, an algorithm receives a poset element one after another. For each new element added, the algorithm takes an irrevocable decision (assigns a color or extends a current representation) just after this element is presented (algorithms that work under such conditions are called on-line). in the second scenario, an algorithm receives an incomparability graph of some poset and a representation of some parts of this graph, and its task is to check whether this partial representation can be extended to a representation of the whole graph that is appropriate for the considered class of graphs. In the context of on-line algorithms, we focus our attention on two problems: partitioning posets into chains, which is a special case of on-line coloring of incomparability graphs, and embedding posets into d-dimentional space Rd. In the context of extending partial representations problems, we are interested in graph classes whose representations introduce a natural order on vertices of these graphs. We focus our attention on:
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27.12.43414 Hanna Palianytsia i Agnieszka Rabiej |
Podstawy Informatyki Chapter 4.5 - 4.9 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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03.03.27101 Aleksandra Mędrek, Marcin Muszalski |
Optymalizacja Kombinatoryczna The Erdős discrepancy problem - Part I |
Erdős discrepancy problem had remained unresolved for more than 80 years. In 2015 Erdős theorem has been proofed by Terrence Tao. We present first part of his proof where he uses a Fourier-analytic reduction obtained as part of the Polymath5 project which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. Terence Tao. The Erdős discrepancy problem. Discrete Analysis. Vol. 2, (2016), pp. 1-20. |
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21.08.24249 Miron Ficek |
Podstawy Informatyki Chapter 4 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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02.07.73645 Wojciech Kruk |
Optymalizacja Kombinatoryczna Randomized Primal-Dual Analysis of RANKING for Online Bipartite Matching |
We give a simple proof that the RANKING algorithm of Karp, Vazirani and Vazirani is 1-1/e competitive for the online bipartite matching problem. The proof is via a randomized primal-dual argument. Primal-dual algorithms have been successfully used for many online algorithm problems, but the dual constraints are always satisfied deterministically. This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. |
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06.08.70907 Bartłomiej Bosek |
Informatyka Teoretyczna A Tight Bound for Shortest Augmenting Paths on Trees |
The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T=(WB, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et. al. that the total length of all shortest augmenting paths found is O(n log n). In this paper we prove a tight O(n log n) upper bound for the total length of shortest augmenting paths for trees improving over O(n log² n) bound.
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21.12.70793 Jakub Czarnowicz |
Podstawy Informatyki Chapter 3 of AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS by Robert Sedgewick, Philippe Flajolet |
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25.02.54480 Sylwester Klocek |
Optymalizacja Kombinatoryczna On-line bipartite matching made simple |
We examine the classic on-line bipartite matching problem studied by Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. Algorithm attempts to match online new vertices with edges. Such a decision, once made, is irrevocable. The objective is to maximize the size of the resulting matching. We will see a sketch of simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 − 1/e. B.E. Birnbaum, C. Mathieu. On-line bipartite matching made simple. SIGACT News 39 (1), 80-87, 2008. |
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15.08.51628 Piotr Wójcik |
Podstawy Informatyki Chapter 4 of Flajolet book "Complex Analysis, Rational and Meromorphic Asymptotic". |
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21.10.35314 Zygmunt Łenyk |
Optymalizacja Kombinatoryczna Handwritten graph diagrams recognition |
| Graph visualisation problem is well known and there are many solutions to it. The reverse process - graph recognition - has been disregarded so far. Such solution has wide applications - from scientific to didactic. This paper focuses on hand-written graphs. Objects do not necessarily have regular shapes and there might be a lot of noise. Using computer vision techniques, we recognize first vertices and then edges. The result of the algorithm is a list of edges and a generated graph visualisation. |
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10.04.32463 Tomasz Kisielewski |
Podstawy Informatyki Logic of Provability by George Boolosa |
| Short presentantion of the book Logic of Provability by George Boolos. |
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15.06.16149 Szymon Borak |
Optymalizacja Kombinatoryczna On some problems in planar graphs |
We give insight into competitive reachability for outerplanar graphs and also for other classes of graphs with bounded degree. Competitive reachability is a game where two players orient the edges of undirected graph G alternately until all edges of G have been oriented. One player wants to minimize the number of ordered pairs of distinct vertices (x, y) with a directed path from x to y. And the second want to maximize it. Furthermore we focus on harmonious coloring conjecture for outerplanar graphs and further attempts in this area. A harmonious coloring of a graph G is a proper vertex coloring of G in which every pair of colors appears on adjacent vertices at most once. The harmonious chromatic number, denoted by h(G), is the minimum number of colors in a harmonious coloring. Analogically we define harmonious edge coloring in which every pair of colors appears on incident edges at most once. The minimal number of color we denote by h'(G). The conjecture states that h(G)<=h'(G). Finally we tackle the hamiltonian cycles in grid graphs. Grid graph are finite vertex induced subsets of infinite lattice, composed from unit-side squares, equilateral triangles or equilateral hexagons. Decide whether the grid graph has hamiltonian cycle is NP-hard in general. |
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12.01.13302 Tomasz Kisielewski |
Podstawy Informatyki Logic of Provability by George Boolosa |
Short presentantion of the book Logic of Provability by George Boolos.
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20.08.59955 Damian Goik |
Informatyka Teoretyczna Succinct progress measures for solving parity games |
The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasipolynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right. Based on the paper:
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14.04.40790 Piotr Wójcik |
Informatyka Teoretyczna On the asymptotic density of valid sentences in first-order logic about one binary relation |
This study arises from the following question: what is the proportion of tautologies of the given length n among the number of all FO relational sentences of length n? We investigate the simplest language with a fixed signature σ = {r}, where r is a binary relation symbol. The model with four logic symbols and an universal quantifier lead us to discover an unexpected result - the fraction of valid sentences is always greater than a fixed constant and therefore the density, if exists, is positive. The main theorem is derived from the analysis of structural properties of FO formulae, which themselves bear strict resemblance to structural properties of λ-terms. |
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14.08.37997 Kamil Sałaś |
Kryptologia Helios: Web-based Open-Audit Voting |
The talk is based on the paper by Ben Adida with the same title [1]. In addition, we recall ElGamal encryption scheme and zero-knwoledge proofs. Voting with cryptographic auditing, sometimes called open-audit voting, has remained, for the most part, a theoretical endeavor. In spite of dozens of fascinating protocols and recent ground-breaking advances in the field, there exist only a handful of specialized implementations that few people have experienced directly. As a result, the benefits of cryptographically audited elections have remained elusive. We present Helios, the first web-based, open-audit voting system. Helios is publicly accessible today: anyone can create and run an election, and any willing observer can audit the entire process. Helios is ideal for on-line software communities, local clubs, student government, and other environments where trustworthy, secret-ballot elections are required but coercion is not a serious concern. With Helios, we hope to expose many to the power of open-audit elections. References [1] Ben Adida, Helios: Web-based Open-Audit Voting, Proceedings of the 17th Conference on Security Symposium, 2008, pp. 335-348 |
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03.06.21515 Jakub Nowak |
Podstawy Informatyki Generic Complexity of Presburger Arithmetic by Alexander N. Rybalov |
| Fischer and Rabin proved in that the decision problem for Presburger Arithmetic has at least double exponential worst-case complexity (for deterministic and nondeterministic Turing machines). In paper 4 a theory of generic-case complexity was developed, where algorithmic problems are studied on "most" inputs instead of set of all inputs. An interesting question rises about existing of more efcient (say, polynomial) generic algorithm deciding Presburger Arithmetic on some set of closed formulas of asymptotic density 1 (so-called generic set). We prove, however, that there is not even an exponential generic algorithm working correctly on a set of inputs which (so-called strongly generic set). |
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29.06.5197 Wojciech Kruk, Piotr Kruk |
Optymalizacja Kombinatoryczna Ulam Sequences and Ulam Sets |
The Ulam sequence is given by a1=1,a2=2, and then, for n≥3, the element an is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence 1,2,3,4,6,8,11,13,16,…, which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors {v1,…,vk}⊂ℝn, we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.
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14.08.87334 Piotr Micek |
Informatyka Teoretyczna Ramsey Theory for Binary Trees and the Separation of Tree-chromatic Number from Path-chromatic Number |
We propose a Ramsey theory for binary trees and prove that for every r-coloring of "strong copies" of a small binary tree in a huge complete binary tree T, we can find a strong copy of a large complete binary tree in T with all small copies monochromatic. As an application, we construct a family of graphs which have tree-chromatic number at most 2 while the path-chromatic number is bounded. This construction resolves a problem posed by Seymour. Joint work with Fidel Barrera-Cruz, Stefan Felsner, Tamás Mészáros, Heather Smith, Libby Taylor, and Tom Trotter. |
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06.02.87225 Grzegorz Bukowiec |
Podstawy Informatyki The Undecidability of the Generalized Collatz Problem by Stuart A. Kurtz and Janos Simon |
| The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. In this paper, we build on work of J. H. Conway to show that a natural generalization of the Collatz problem is $PI^0_2$ complete. |
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14.12.84541 Jan Derbisz |
Kryptologia Subquadratic Greatest Common Divisor |
| The binary algorithm is a variant of the Euclidean algorithm that performs well in practice. We present a quasi-linear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modified version of the binary algorithm. The structure of the algorithm is very close to the one of the well-known Knuth-Schonhage fast gcd algorithm; although it does not improve on its O(M(n) log n) complexity, the description and the proof of correctness are significantly simpler. This leads to a simplification of the implementation and to better running times. |
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06.03.70907 Sylwester Klocek, Maciej Woźniak |
Optymalizacja Kombinatoryczna On the complexity of the chip-firing reachability problem |
| In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. Both of these algorithms are strongly polynomial. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraph |
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02.10.68059 Piotr Wójcik |
Podstawy Informatyki Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications by O.Bodini, J. David, and Ph. Marchal |
| In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a multiplicative constant. We then apply a natural process “simulate-guess-and-proof” to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height phenomenon. |
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08.08.65376 Szymon Policht |
Kryptologia Supersingular isogeny key exchange |
| Supersingular isogeny is the newest addition to the post-quantum cryptography roster. It is elliptic curve based, but unlike tradidional ECC algorithms, it's quantum resistant. It offers significant key size reduction and computation time speedup compared to other post-quantum algorithms. |
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29.10.51741 Katrzyna Janocha |
Optymalizacja Kombinatoryczna Proper Orientations of Planar Bipartite Graphs |
| An orientation of a graph G is proper if any two adjacent vertices have different indegrees. The proper orientation number χ (G) of a graph G is the minimum of the maximum indegree, taken over all proper orientations of G. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that χ (G) ≤ 4, when G is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound. |
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23.06.32576 Anna Kobak |
Optymalizacja Kombinatoryczna Lambda number for the direct product of some family of graphs |
| An L(2,1) labeling for a graph G = (V,E) is a function f on V such that | f(u) - f(v)| >= 2 if u,v are adjacent and f(u), f(v) are distinct if u,v are vertices of distance two. The lambda(G) for G is the minimum span over all L(2,1) labelings of G. We will show that when m>=6 and n>=3, lambda(Pm x Cn) = 7 if and only if n is not a multiple of 7 and also provide the conditions on m and n such that lambda(Cm x Cn) <= 7. |
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28.07.29838 Torsten Ueckerdt Karlsruhe Institute of Technology |
Informatyka Teoretyczna The k-Strong Induced Arboricity of a Graph |
Motivated by a connection to vertex-distinguishing colorings, we initiate the study of a new graph covering parameters: The k-strong induced arboricity. For a graph G and a positive integer k, a k-strong induced forest F in G is an induced forest in which every component has at least k edges. An edge in G is called k-valid if it is contained in at least one k-strong induced forest. The k-strong induced arboricity fk(G) is the smallest number m such that all k-valid edges of G can be covered with m k-strong induced forests in G. |
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20.01.29729 Maciej Bendkowski |
Podstawy Informatyki Analytic combinatorics: an introduction |
In our talk we outline the main concepts and techniques of analytic combinatorics used to investigate properties of large random algebraic structures. We discuss the central interpretation of generating functions as functions analytic at the origin allowing to relate their analytic properties with the quantitative properties of studied structures. Finally, we briefly excerpt the techniques of singularity analysis allowing us to access the asymptotic form of corresponding counting sequences or investigate the probability distribution of interesting combinatorial parameters.
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27.11.27045 Aleksandra Nowak |
Kryptologia The Fully Homomorphic Encryption and Approximate Greatest Common Divisor Problem |
We briefly introduce the definition of fully homomorphic encryption and describe the two main problems on which are based latest FHE schemes: The LWE/Ring-LWE and AGCD problems. We discuss their advantages and the relations between them. We present the definition of bootstrapping and investigate the FHE scheme based on the AGCD problem as published in [1]. References [1] J. H. Cheon, D. Stehlé, Fully Homomorphic Encryption over the Integers Revisited, EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques, pp. 24--43. |
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17.02.13411 Grzegorz Bukowiec |
Optymalizacja Kombinatoryczna Even factors of graphs |
| An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. It has been shown that if a graph G has an even factor, it also has an even factor F such that |E(F)| >= 4/7 (|E(G)| + 1). 4/7 is the best possible ratio here, but we will try to strengthen this lower bound by taking the set of vertices of degree 2 into consideration. |
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28.11.76382 Jakub Szarawski |
Optymalizacja Kombinatoryczna A greedy approach to the Turtle Tower problem |
| In the Turtle Tower problem we are given n turtles with a mass and capacity for each of them. We are looking for the highest tower possible, regarding that capacity of every turtle in the tower cannot be exeeded by the sum of the masses of turles it carry. Presented solution is faster than commonly known dynamic one. |
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31.12.73644 Marcin Pilipczuk University of Warsaw |
Informatyka Teoretyczna Subexponential Parameterized Algorithms for Planar Graphs, Apex-Minor-Free Graphs and Graphs of Polynomial Growth via Low Treewidth Pattern Covering |
We prove the following theorem. Given a planar graph G and an integer k, it is possible in polynomial time to randomly sample a subset A of vertices of G with the following properties: 1) A induces a subgraph of G of treewidth 2) for every connected subgraph H of G on at most k vertices, the probability that A covers the whole vertex set of H is at least Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound In the talk I will first focus on the background and motivation, and then highlight the main ideas of the proof by sketching the proof for the case of graph classes of polynomial growth. Based on joint work with Fedor Fomin, Daniel Lokshtanov, Dániel Marx, Michał Pilipczuk, and Saket Saurabh: http://arxiv.org/abs/1604.05999 and http://arxiv.org/abs/1610.07778. |
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27.06.73535 Konrad Kalita |
Podstawy Informatyki Java Generics are Turing Complete by Radu Grigore |
| This paper describes a reduction from the halting problem of Turing machines to subtype checking in Java. It follows that subtype checking in Java is undecidable, which answers a question posed by Kennedy and Pierce in 2007. It also follows that Java’s type checker can recognize any recursive language, which improves a result of Gil and Levy from 2016. The latter point is illustrated by a parser generator for fluent interfaces. |
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02.05.70852 Michał Ziobro |
Kryptologia Introduction to Homomorphic Encryption |
The talk is divided into two parts. In the first part we briefly introduce Fully Homomorphic Encryption and a presentation of a classic example described in [1]. In the second part, we bring up a subject of partially homomorphic encrytpion schemes over finite fields, presented in [2]. References: [1] C. Gentry, Computing Arbitrary Functions of Encrypted Data, 2008 (pdf) |
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23.07.57217 Helena Borak, Zygmunt Łenyk |
Optymalizacja Kombinatoryczna Necklaces, Convolutions, and X + Y, A new upper bound for the online square packing |
| Necklaces, Convolutions, and X + Y The necklace alignment problem is to find the optimal rotation of the necklaces to best align the beads, when we have two necklaces given, each with n beads at arbitrary positions. Alignment is measured according to the ℓ_p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = ∞ and how these problems can be reduced to convolution problems which can be solve in subquadratic time. Besides, we say how the necklace alignment problems, and their corresponding convolution problems, are also intrinsically connected to problems on X + Y matrices. A new upper bound for the online square packing In online square packing problem we try to minimise the height of squares on a plane with width 1. Squares come one by one, they can’t overlap and once set, it’s position can’t be changed. A new upper bound (ratio between algorithm result and optimal packing) is found by applying modified version of previously used First Fit Shelf algorithm. |
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26.08.54479 Lech Duraj, Adam Polak |
Informatyka Teoretyczna Longest Common Strictly Increasing Subsequecnce |
The Longest Common Increasing Subsequence problem is a variant of classic Longest Common Subsequence problem, which can be solved in quadratic time with a simple dynamic programming algorithm. During the talk we will show a reduction from the Orthogonal Vectors problem to the Longest Common Increasing Subsequence problem which proves that the latter cannot be solved in strongly subquadratic time unless the SETH is false.
Simple modifications of the reduction prove that the problem for k sequences cannot be solved in O(nk-ε) time, that the same lower bounds apply to the Longest Common Weakly Increasing Subsequence, and that the assumption of SETH can be replaced with a weaker statement about satisfiability of non-deterministic branching programs. |
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14.10.35204 Jarek Duda |
Podstawy Informatyki Boundaries for hashing problem, or how many bits ones individuality costs |
| I will talk about information-theoretic boundaries for the hashing problem, the Bloom filter, and generally about informational content of structures like trees and graphs. While the label size has to grow like logarithm of the population size, neglecting information about the order (lg(n!) bits), we get a linear growth of entropy of population, allowing to bound 'the cost of individuality' asymptotically to ~2.33275 bits per object. |
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10.11.18886 Andrzej Głuszyński, Jakub Nowak |
Optymalizacja Kombinatoryczna Local Antimagic Vertex Coloring of a Graph, A short proof of Cayley's tree formula |
| Local Antimagic Vertex Coloring of a Graph The edge labelling is called 'local antimagic', if for all vertices sum of labels for incident edges is different for every two adjacent vertices. Such sum induce a correct vertex colouring. The local antimagic chromatic number - X_la(G) - is the minimum number of colours used by any proper local antimagic labelling. In the paper authors present results on this parameter for trees, friendship, wheel and clique graphs. A short proof of Cayley's tree formula Cayley’s tree formula is a very elegant result in Graph Theory. The problem is to find the number of all possible trees on a given set of labeled vertices. For n = 2 and vertex set {v1, v2}, we have only one tree. For n = 3 and vertex set {v1, v2, v3}, we have 3 different trees. Similarly for n = 4, we have 16 trees. We give a short proof of Cayley’s tree formula for counting the number of different labeled trees on n vertices. Alok Bhushan Shukla, A short proof of Cayley's tree formula. |
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09.06.16039 Szymon Stankiewicz |
Podstawy Informatyki CANTOR POLYNOMIALS AND THE FUETER-POLYA THEOREM by MELVYN NATHANSON |
A packing polynomial is a polynomial that maps the set N^2 of lattice points with nonnegative coordinates bijectively onto N. Cantor constructed two quadratic packing polynomials, and Fueter and Polya proved analytically that the Cantor polynomials are the only quadratic packing polynomials. |
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15.04.13356 Mateusz Jachna |
Kryptologia Secure Hash Algorithms family and the recently found collision for SHA-1 |
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21.12.79065 Piotr Wójcik |
Kryptologia Quantum Authentication with Key Recycling |
We show that a family of quantum authentication protocols introduced in FOCS 2002 can be used to construct a secure quantum channel and additionally recycle all of the secret key if the message is successfully authenticated, and recycle part of the key if tampering is detected. We give a full security proof that constructs the secure channel given only insecure noisy channels and a shared secret key. We also prove that the number of recycled key bits is optimal for this family of protocols, i.e., there exists an adversarial strategy to obtain all non-recycled bits. Previous works recycled less key and only gave partial security proofs, since they did not consider all possible distinguishers (environments) that may be used to distinguish the real setting from the ideal secure quantum channel and secret key resource. References: [1] Christopher Portmann, Quantum Authentication with Key Recycling (pdf) |
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13.03.65431 Aleksandra Mędrek, Marcin Muszalski |
Optymalizacja Kombinatoryczna Planning for Fast Connectivity Updates |
| Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d > 1 deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linear-space representation of graphs which enables us to determine how a batch of edge updates can impact the graph. Given a set of d edge updates, in time O(d polylg n) we can obtain the number of connected components, the size of each component, and a fast oracle for answering connectivity queries in the updated graph. The initial representation is polynomial-time constructible. |
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16.08.59900 Jan Szczepaniec |
Kryptologia Inclusive Block Chain Protocols |
Distributed cryptographic protocols such as Bitcoin and Ethereum use the block chain to synchronize a global log of events between nodes in their network. Previous research has shown that the mechanics of the block chain and block propagation are constrained: if blocks are created at a high rate compared to their propagation time in the network, many conflicting blocks are created and performance suffers greatly.
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05.11.46265 Patryk Urbański |
Optymalizacja Kombinatoryczna Generating Linear Extensions Fast |
One of the most important sets associated with a poset P is its set of linear extensions, E(P). In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)), where e(P) = |E(P)|. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n*e(P)), where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y), for all pairs x,y, in time O(n^2 + e(P)). |
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02.02.46211 Jakub Cisło, Grzegorz Jurdziński |
Tight Hardness Results for LCS and other Sequence Similarity Measures |
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10.12.43527 Manuel Bodirsky TU Dresden |
Informatyka Teoretyczna The tractability conjecture for finitely bounded homogeneous structures |
| Finitely bounded homogeneous structures form a large class of infinite structures that can be seen as a generalisation of the class of all finite structures. Many results about finite structures, in particular in the context of the complexity of constraint satisfaction problems, can be generalised to this larger class. In this talk I will present a reformulation of a tractability conjecture for CSPs for this class in terms of polymorphisms, due to Barto and Pinsker (LICS 2016), and I will present a proof that the condition given in the tractability conjecture is decidable (under some Ramsey-theoretic assumptions that might hold in general for all reducts of finitely bounded homogeneous structures). |
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04.06.43418 Łukasz Lachowski |
Podstawy Informatyki Impossibility of Distributed Consensus with One Faulty Process by MICHAEL J. FISCHER, NANCY A. LYNCH AND MICHAEL S. PATERSO |
| The consensus problem involves a asynchronous system of processes, some of which may be unreliable.The problem is for the reliable processes to agree on a binary value. In this paper, it is shown that every protocol for this problem has the possibility of nontermination, even with only one faulty process. By way of contrast, solutions are known for the synchronous case, the “Byzantine Generals” problem. |
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11.04.40735 Marcin Briański |
Kryptologia Non-Interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers |
The talk is based on the paper with the same title by Rosario Gennaro, Craig Gentry and Bryan Parno. Verifiable Computation enables a computationally weak client to "outsource" the computation of a function F on various inputs x1, ..., xk to one or more workers. The workers return the result of the function evaluation, e.g., yi = F(xi), as well as a proof that the computation of F was carried out correctly on the given value xi. The verification of the proof should require substantially less computational effort than computing F(xi) from scratch. We present a protocol that allows the worker to return a computationally sound, non-interactive proof that can be verified in O(m) time, where m is the bit-length of the output of F. The protocol requires a one-time pre-processing stage by the client which takes O(|C|) time, where C is the smallest Boolean circuit computing F. Our scheme also provides input and output privacy for the client, meaning that the workers do not learn any information about the values xi or yi. |
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01.07.27100 Grzegorz Matecki |
Optymalizacja Kombinatoryczna Boolean dimension of posets |
| A boolean dimension bdim(P) of a poset P=(X,<) is a smallest number k for which there is a set l1, l2, ..., lk of labelings X:->N and a boolean formula f(a1, ..., ak) such that the following is true: x < y in P iff f(a1, .., a_k) = 1 where ai =1 iff li(x) < li(x). Generally, it is simple to observe that bdim(P) <= dim(P). Also, it is known that there is a constant c such that bdim(n) <= c log(n) for any poset P of size n. The are few interesting open problems for boolean dimension: 1) Is boolean dimension of the boolean lattice of size n less that n? 2) Is there a constant c such that bdim(P) < c for any planar poset P? |
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19.08.27041 Sylwester Klocek, Wojciech Kruk |
The Alternating Stock Size Problem and the Gasoline Puzzle |
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27.01.24253 Maciej Bendkowski |
Podstawy Informatyki Boltzmann samplers: random generation of combinatorial structures with an application to lambda calculus |
| In their seminal paper, Duchon et al. proposed a surprisingly simple, general-purpose framework of Boltzmann samplers meant for random generation of combinatorial structures. In this talk we revisit their method and discuss its elegant relation with analytic combinatorics as well as important practical implementation details. Finally, we discuss the application of Boltzmann samplers to the random generation of lambda terms used, e.g. in testing functional programming compilers. |
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04.12.21569 Zygmunt Łenyk |
Kryptologia Speeding up modular multiplication using Montgomery and Barrett reduction |
| In the talk we present Montgomery and Barrett reductions that are used to speed up modular computations. In both reductions some pre-computations are made allowing for replacing subsequent expensive divisions by some fixed modulus with much cheaper operations involving a suitable power of 2. This is particularly useful when many modular divisions by the same modulus are performed (for example in finite field arithmetic or in RSA). |
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04.05.2076 Mateusz Twaróg, Łukasz Majcher |
Optymalizacja Kombinatoryczna Combinatorial library core |
| Presentation and discussion on core functionalities of the c++ combinatorial library. introduction to classes representing graphs, graph traversing algorithm templates and simple GUI. |
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22.09.5087 Michał Zwonek |
Podstawy Informatyki Wielomianowe kodowania |
Rozważany będzie problem istnienia wielomianowej bijekcji, najniższego stopnia, między N^k, a N. Przedstawione będą także problemy otwarte związane z tą tematyką. Materiały do wystąpienia: 1) Elementarny dowód Twierdzenie Feuter-Polya (jedyny kwadratowy i bijektywny wielomian N^2->N to funkcja cantore'a) https://arxiv.org/abs/1512. 2) Praca, w której autorzy pokazują nieistnienie wielomianów 3 i 4 stopnia. http://www.sciencedirect.com/ 3) Praca podobnie tematyczna odnosząca się do problemu istnienia wielomianów bijektywnych z pewnego sektora N^2 w N. (To o czym wspomniałem na koniec, opis tego problemu jest też pod koniec w 1) ). Pod koniec pracy jest opisane 6 problemów otwartych związanych z tą tematyką. https://arxiv.org/abs/1305. 4) W podobnej tematyce. http://www.sciencedirect.com/science/article/pii/0022314X78900355
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28.02.2017 Michał Dyrek |
Kryptologia LLL algorithm and its applications in Number Theory and Cryptography |
| The talk is devoted to the algorithm by A. Lenstra, H. Lenstra and L. Lovász dated 1982 allowing for approximation of Shortest Vector Problem in polynomial time. We will present the idea of the algorithm and highlight its applications such as factoring polynomials over Q, constructing polynomials with small coefficients and connections with attacks on RSA. |
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02.10.73644 Wojciech Kruk, Maciej Woźniak |
Optymalizacja Kombinatoryczna A few open problems |
| We mentioned the following open problems in graph theory and discrepancy theory: 1. Erdos discrepancy problem 2. Hoang - Reed conjecture 3. Seagull problem - a consequence of Hadwiger's conjecture |
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06.11.70906 29.03.5197 Grzegorz Guśpiel |
Informatyka Teoretyczna Partial Visibility Representation Extension Problem |
We study a class of graphs that have a special geometric representation. By a bar visibility representation of an undirected graph we mean a function that associates with each vertex of a graph a horizontal line segment in such a way, that between segments representing two ends of an edge there is a vertical strip (of visibility). In case of directed graphs, we additionally assume that the visibility is from the bottom to the top, that is the line segment representing the source of the edge is below the one for the target. Graphs admitting such representations are well understood and can be recognized in linear time, both in the undirected and in the directed case. We work in a more subtle setting, where line segments are already associated with some vertices of a graph, and the question is if this can be extended to a bar visibility representation of an entire graph. We prove some results on complexity of this kind of problems. This is joint work with Steven Chaplick, Grzegorz Gutowski, Tomasz Krawczyk and Giuseppe Liotta. The manuscript is available here: https://arxiv.org/abs/1512.00174 |
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23.03.70793 Sylwester Klocek |
Podstawy Informatyki Incompleteness, Undecidability and Automated Proofs by Cristian S. Calude and Declan Thompson |
| Incompleteness and undecidability have been used for many years as arguments against automatising the practice of mathematics. The advent of powerful computers and proof-assistants – programs that assist the development of formal proofs by human-machine collaboration – has revived the interest in formal proofs and diminished considerably the value of these arguments. In this paper we discuss some challenges proof-assistants face in handling undecidable problems – the very results cited above – using for illustrations the generic proof-assistant Isabelle. |
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24.01.2017 Kamil Sałaś |
Kryptologia Lower Bounds for Discrete Logarithms |
| In the talk we will present the computational complexity of the discrete logarithm in the context of "generic algorithms", that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as unique binary string. For discrete logarithm, any generic algorithm must perform Ω(p^1/2) group operations, where p is the largest prime dividing the order of the group. |
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19.01.2017 Paweł Petecki Akademia Górniczo-Hutnicza |
Optymalizacja Kombinatoryczna Symmetry breaking polynomial |
| Let G be a graph, and let Γ= Aut G. A coloring c of G is symmetry-breaking if for every non-identity automorphism φ in Γ, there is some vertex v of G such that c(v)≠c(φ(v)). There has been a lot of work on the minimum number of colors in a symmetry-breaking coloring of G. We discuss here a different problem: counting the number of k-colorings that are symmetry breaking. The tool, as is frequently the case for problems such as this one, is Möbius inversion. To solve this problem we define symmetry breaking polynomial ψG. For positive integer k (number of colors), ψG(k) is the number of k-colorings that break all non-trivial symmetries of the graph G. |
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01.07.51741 Marian Mrozek |
Informatyka Teoretyczna The discrete charm of Morse theory |
| The lecture will start with recalling P.S. Alexandroff's Theorem (1937) on mutual equivalence of posets and T0 topologies on finite sets. Next, we will outline the combinatorial version of the classical Morse Theory presented by R. Forman in 1998. Then, we will elaborate Forman's ideas towards the combinatorial topological dynamics with potential applications in Big Data problems and time series. The topics of the lecture will be expanded in a course for PhD students in the summer semester 2016/17. |
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16.11.51627 Michał Ziobro |
Podstawy Informatyki Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search by Jose Espırito Santo, Ralph Matthes, Luıs Pinto |
| Kontynuacja seminarium z 23.11.2016 |
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17.01.2017 Grzegorz Bukowiec |
Kryptologia A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic |
Until recently, all the algorithms for computing discrete logarithm had a sub-exponential complexity of type L(1/3), similar to the factorization problem. In this talk we'll discuss a heuristic algorithm that provides quasi-polynomial complexity for discrete logarithm in finite fields of small characteristic and that even for other cases still surpasses the Function Field Sieve method. References: [1] R. Barbulescu, P. Gaudry, A. Joux, E. Thomé, A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic (pdf) |
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24.02.32576 Patryk Mikos |
Informatyka Teoretyczna Online coloring of intervals with bandwidth |
We study the online interval coloring problem with bandwidth. The input is a sequence of pairs Ji= (Ii,wi) where Ii is an interval on the real line and wi is a real number from (0,1]. In this setting a proper coloring is a function f:Ji →N such that for each color c and any point p on the real line, the sum of bandwidths of intervals containing p and colored by c does not exceed 1. The best known lower bound on the competitive ratio in this problem is 24/7. We present an explicit strategy for Presenter that increases the competitive ratio ifor this problem to at least 4.1626. |
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11.07.32462 Patryk Mikos |
Podstawy Informatyki ON THE NUMBER OF DISTINCT LANGUAGES ACCEPTED BY FINITE AUTOMATA WITH n STATES by Michael Domaratzki, Derek Kisman and Jeffrey Shallit |
| We give asymptotic estimates and some explicit computations for both the number of distinct languages and the number of distinct finite languages over a k-letter alphabet that are accepted by deterministic finite automata (resp. nondeterministic finite automata) with n states. |
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10.01.2017 Szymon Policht |
Kryptologia Faster operations on elliptic curves using Edwards curves |
| Elliptic curve cryptography is a broad and commonly used section of modern-day cryptography. Because of that, the speed of elliptic curve operations directly impacts the performance of many current systems. In this talk we'll show how to speed up those operations using Edwards curves. References: [1] Bernstein D.J., Lange T. (2007) Faster Addition and Doubling on Elliptic Curves. In: Kurosawa K. (eds) Advances in Cryptology – ASIACRYPT 2007. ASIACRYPT 2007. Lecture Notes in Computer Science, vol 4833. Springer, Berlin, Heidelberg (https://eprint.iacr.org/2007/286.pdf) |
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12.12.16093 Jan Derbisz, Jakub Łabaj |
Sortowanie przez spacer po drzewie |
Rozważamy następujący problem: wierzchołki drzewa ponumerowane są kolejnymi liczbami naturalnymi, a dodatkowo w wierzchołku x leży skrzynka o numerze p(x), przy czym funkcja p jest permutacją zbioru {1,2,..,n}. Rozważamy chodzącego po drzewie robota, który może w danym momencie trzymać tylko jedną skrzynkę, może też podnieść napotkaną skrzynkę upuszczając aktualnie trzymaną. Celem robota jest posortować skrzynki (przenosząc każdą do wierzchołka o odpowiednim numerze), przechodząc po drzewie najkrótszą możliwą ścieżką. Praca D. Grafa podaje algorytm znajdujący taką ścieżkę w czasie O(n2) oraz dowód, że jeśli drzewo zastąpimy grafem planarnym, problem staje się NP-zupełny. |
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05.03.13297 Konrad Kalita |
Podstawy Informatyki ANALYTIC MODELS AND AMBIGUITY OF CONTEXT-FREE LANGUAGES by Philippe Flajolet |
| We establish that several classical context-free languages are inherently ambiguous by proving that their counting generating functions, when considered as analytic functions, exhibit some characteristic form of transcendental behaviour. To that purpose, we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages In particular, the paper contains a general density theorem for unambiguous context-free languages. |
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15.01.62693 Łukasz Majcher, Jan Szczepaniec |
Optymalizacja Kombinatoryczna Convex p-partitions of bipartite graphs |
| A set of vertices X of a graph G is convex if no shortest path between two vertices in X contains a vertex outside X. We prove that for fixed p ≥ 1, all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time. |
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19.02.59955 Maciej Bendkowski |
Informatyka Teoretyczna Boltzmann samplers: a framework for random generation of combinatorial structures with an application to lambda calculus |
| In their seminal paper, Duchon et al. proposed a surprisingly simple, general-purpose framework of Boltzmann samplers meant for random generation of combinatorial structures. In this talk we revisit their method and discuss its elegant relation with analytic combinatorics as well as important practical implementation details. Finally, we discuss the application of Boltzmann samplers to the random generation of lambda terms used, e.g. in testing functional programming compilers. |
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07.12.43472 Michał Glapa, Franciszek Stokowacki |
Skojarzenia w grafach metodami algebraicznymi |
| Na seminarium omówiony zostanie przełomowa praca z 2006, autorstwa Muchy i Sankowskiego, opisująca algorytm obliczania skojarzeń w grafach za pomocą eliminacji Gaussa. Przedstawiony algorytm ma złożoność zależną od mnożenia macierzy, niższą niż O(n2.5), algorytmu Micaliego-Vaziraniego, który bardzo długo był najlepszą znaną metodą. |
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15.12.2016 Anna Kobak |
Optymalizacja Kombinatoryczna Open problems in graph theory concerning minors. |
We mentioned following open problems in graph theory:
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14.10.40789 Grzegorz Matecki |
Informatyka Teoretyczna Two-Dimensional Irregular Packing Problem |
| We present results on packing irregular shapes onto given sheets of material. |
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29.02.40676 Piotr Wójcik |
Podstawy Informatyki Enumeration and random generation of accessible automata by Frederique Bassino and Cyril Nicaud |
| We present a bijection between the A_n of deterministic and accessible automata with n states on a k-letters alphabet and some diagrams, which can themselves be represented as partitions of a set of kn + 1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of A_n is related to the Stirling number. Our bijective approach also yields an efficient random sampler, for the uniform distribution, of automata with n states, its complexity is O(n^3/2), using the framework of Boltzmann samplers. |
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04.04.37938 Krzysztof Kleiner |
Kryptologia An introduction to quantum computing and cryptography I |
In this talk we're going to discuss quantum informatics and its impact on the field of cryptography. We will introduce the basic concepts of quantum computing as well as cryptography based on Quantum Key Distribution scheme, one of the aspects of quantum informatics which already is being used in practice. Then we will present Shor's algorithm for polynomial-time factorization, responsible for the cryptosystems based on the hardness of factorization or discrete logarithm (in abelian groups) being no longer secure against an adversary with access to a quantum computer.
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08.12.2016 Lech Duraj |
Krótka opowieść o mnożeniu macierzy |
W ostatnich latach pojawiają się kolejne, coraz lepsze algorytmy mnożenia macierzy. Każdy z nich jest jednak tylko nieznacznie szybszy od poprzednich, będąc przy tym nierównie trudniejszy w zrozumieniu i analizie. Fakt ten jeszcze bardziej komplikuje otwarte od wielu lat pytanie o złożoność optymalnego algorytmu mnożenia macierzy. Celem prezentacji jest krótkie omówienie technik używanych do ataków na ten niezwykle ważny i trudny problem. Prezentacja oparta jest na przeglądowym wykładzie François Le Galla (autora ostatnich wyników w tym temacie) z 2014 roku. |
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08.12.2016 Zygmunt Łenyk |
Optymalizacja Kombinatoryczna Rendezvous on the line. |
| This is one of a handful of rendezvous problems where two players must find one another in a certain structured domain. In line case, players are placed on the line with distance 2, without knowing neither on which side is their partner, nor the direction of the line. I'll concentrate on the symmetric case where players must follow a specific (but maybe mixed) strategy. The conjecture is that best expected time of meeting two players equals 4.25. |
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25.10.21510 Jakub Brzeski |
Podstawy Informatyki ENUMERATION OF FORMAL LANGUAGES by Michael Domaratzki |
| We survey recent results on the enumeration of formal languages. In particular, we consider enumeration of regular languages accepted by deterministic and nondeterministic finite automata with n states, regular languages generated by regular expressions of a fixed length, and !-regular languages accepted by Müller automata. We also survey the uncomputability of enumeration of context-free languages and more general structures. |
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06.12.2016 Marek Rusinowski |
Kryptologia Security of instant messaging applications. |
| Nowadays billions of people around the world are sharing sensitive information using instant messaging applications. We will look into the current state of IM security, the problems in this area and a few encryption protocols---OTR and Signal Protocol in particular---that provide security features desired by users. |
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01.12.2016 Aleksandra Mędrek, Krzysztof Maziarz |
Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back |
Praca Aleksandra Mądrego opisuje nowe podejście do problemu maksymalnego przepływu, z użyciem tzw. przepływów elektrycznych. W tej technice krawędziom przypisywany jest opór, a zadaniem jest zminimalizowanie wydzielonej energii. Dowolną sieć przepływową można zredukować do zadania przepływu elektrycznego, z użyciem pośredniej redukcji poprzez warianty problemu skojarzenia w grafie dwudzielnym. Głównym rezultatem pracy jest algorytm przepływu o złożoności O(m10/7), na seminarium będzie prezentowana prostsza wersja algorytmu, działająca w O(m3/2). |
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01.12.2016 Patryk Urbański |
Optymalizacja Kombinatoryczna Coloring Ordinary Maps, Maps of Empires and Maps of the Moon |
A short review of generalized map coloring problems:
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01.12.2016 Mateusz Twaróg |
Optymalizacja Kombinatoryczna Second Neighborhood via First Neighborhood in Digraphs |
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22.12.84650 Bartosz Walczak |
Informatyka Teoretyczna Coloring curves that cross a fixed curve |
| A class of graphs is χ-bounded if the chromatic number of all graphs in the class is bounded by some function of their clique number. String graphs are intersection graphs of curves in the plane. Significant research in combinatorial geometry has been devoted to understanding the classes of string graphs that are χ-bounded. In particular, it is known since 2012 that the class of all string graphs is not χ-bounded. We prove that for every integer t≥1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is χ-bounded. This result is best possible in several aspects; for example, the upper bound t on the number of crossings with the fixed curve cannot be dropped. As a corollary, we obtain new upper bounds on the number of edges in so-called k-quasi-planar topological graphs. This is joint work with Alexandre Rok. |
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04.08.84482 Yauheni Ananchuk |
Podstawy Informatyki ALGEBRAIC FOUNDATIONS FOR QUALITATIVE CALCULI AND NETWORKS by ROBIN HIRSCH, MARCEL JACKSON, AND TOMASZ KOWALSKI |
| Binary Constraint Problems have traditionally been considered as Network Satisfaction Problems over some relation algebra. A constraint network is satisfable if its nodes can be mapped into some representation of the relation algebra in such a way that the constraints are preserved. A qualitative representation is like an ordinary representation, but instead of requiring (a ; b) = a j b , as we do for ordinary representations, we only require that. A constraint network is qualitatively satisfable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfable then it is clearly qualitatively satisfable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfable if and only if it is qualitatively satisfable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infnite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always ; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebra |
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29.11.2016 Anna Kobak |
Kryptologia Breaking RSA vs Factoring in generic ring model |
In the talk we present results of Aggarwal and Maurer [1], who showed that a generic ring algorithm for breaking RSA with modulus $N$ can be converted into an algorithm for factoring $N$. The results imply that any attempt at breaking RSA without factoring $N$ will be non-generic and hence will have to manipulate the particular bit-representation of the input modulo $N$. This provides new evidence that breaking RSA may be equivalent to factoring the modulus.
References: [1] D. Aggarwal, U. Maurer, Breaking RSA Generically is Equivalent to Factoring, EUROCRYPT 2009 |
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24.11.2016 Wojciech Łopata |
Optymalizacja Kombinatoryczna Several open problems from game theory, graph theory and combinatorics. |
I'll briefly introduce the audience to two unrelated areas: book embedding and mechanism design, and walk through some open problems in those areas. |
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24.11.2016 Dominika Salawa, Jakub Cisło |
Greedy algorithms for Steiner forest |
Referowana praca rozstrzyga długo otwarty problem: czy budowanie drzewa Steinera zachłannym algorytmem daje wynik gorszy od optymalnego o stałą multiplikatywną? Autorzy (A. Gupta, A. Kumar) dowodzą, że tak, dla stałej równej 96. Jest to pierwsze znane oszacowanie wyniku algorytmu zachłannego, wcześniej podawane algorytmy aproksymacyjne dla tego problemu oparte były o programowanie liniowe. |
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23.11.2016 Piotr Danilewski |
Informatyka Teoretyczna Functional Code Building |
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23.11.2016 Michał Ziobro |
Podstawy Informatyki Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search by Jos´e Espırito Santo, Ralph Matthes, Luıs Pinto |
| A new, comprehensive approach to inhabitation problems in simply-typed lambda-calculus is shown, dealing with both decision and counting problems. This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently. The representation may be seen as extending the Curry-Howard representation of proofs by lambda-terms, staying within the methods of lambda-calculus and type systems. Our methodology reveals inductive descriptions of the decision problems, driven by the syntax of the proof-search expressions, and the end products are simple, recursive decision procedures and counting functions. |
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17.11.2016 Patryk Gołębiowski, Wojciech Kruk |
Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms |
| Tematem referatu jest praca Colina White'a dotycząca algorytmów poszukiwania ścieżki na grafach o szczególnym własnościach, mających w założeniu modelować rzeczywiste sieci dróg. Autor analizuje najpopularniejsze istniejące algorytmy i podaje dolne ograniczenia na ich złożoność. |
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16.11.2016 Bartłomiej Bosek |
Informatyka Teoretyczna Every digraph is majority 4-choosable |
| A majority coloring of a digraph is a coloring of its vertices such that for each vertex at most half of its out-neighbors has the same color as that vertex. A digraph D is majority k-choosable if for any assignment of color lists of size k to the vertices there is a majority coloring of D from these lists. We prove the statement in the title. This gives a positive answer to a question posed recently in 1. This is a joint work with Marcin Anholcer and Jarosław Grytczuk. |
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16.11.2016 Michał Zieliński |
Podstawy Informatyki Most programs stop quickly or never halt by Cristian S. Calude and Michael A. Stay |
| The aim of this paper is to provide a probabilistic, but non-quantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N-bit program has halted by a random time.We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k >0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{−k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long” runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density. |
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15.11.2016 Piotr Kawałek |
Kryptologia Teoretyczne podstawy kryptoanalizy |
| Celem referatu jest przedstawienie teoretycznych modeli ataków kryptoanalitycznych oraz tematów pokrewnych wraz z przykładami. |
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10.11.2016 Magdalena Gargas, Mateusz Jachna |
Max flows in O(nm) time, or better |
| W pracy opisany jest nowy algorytm przepływu działający w czasie O(nm + m16/15 log2 n). Istotny jest fakt, że przez połączenie go z poprzednio znanymi algorytmami daje to pozytywną odpowiedź na pytanie, czy da się obliczyć maksymalny przepływ w czasie O(nm). Autorem pracy jest James B. Orlin. |
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09.11.2016 26.10.2016 Adam Polak |
Informatyka Teoretyczna Open problems in on-line and approximation algorithms |
During the talk I will present several promising open problems including:
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09.11.2016 Wojciech Kruk |
Podstawy Informatyki On the generic undecidability of the Halting Problem for normalized Turing machines by Alexander Rybalov |
| Hamkins and Miasnikov presented in 2004 a generic algorithm deciding the classical Halting Problem for Turing machines with one-way tape on a set of asymptotic probability one (on a so-called generic set). Rybalov in 2007 showed that there is no generic algorithm deciding this problem on strongly generic sets of inputs (some subclass of generic sets). In this paper we prove that there is no generic algorithm deciding the Halting Problem for normalized Turing machines on generic sets of inputs. Normalized Turing machines have programs with the following natural restriction: internal states with large indices are not allowed at the beginning of the program. This condition does not reduce the class of computable functions because for every Turing machine there exists a normalized Turing machine which computes the same function. Our proof holds for machines with one-way and two-way tape. It also implies that the Hamkins-Miasnikov algorithm is not generic for normalized Turing machines. |
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08.11.2016 Patryk Gołębiowski |
Kryptologia Advanced Encryption Standard |
| Advanced Encryption Standard (AES) is one of the most popular and widely adopted symmetric encryption scheme. In the talk we discuss how it works and why it is considered safe by the U.S. National Institute of Standards and Technology to use it for protecting classified information. |
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03.11.2016 Gabriel Jakóbczak |
Optymalizacja Kombinatoryczna Proper orientations of some types of graphs |
| Let G be a simple graph. We say that orientation of graph G is proper if for every pair of adjacent veritces u and v their indegrees are different. It was proved by Mieczysław Borowiecki, Jarosław Grytczuk and Monika Pilśniak that for every simple graph exists its proper orientation. Now we can define the proper orientation number of graph G as the minimum of the maximum indegree, taken over all proper orientations of G. We show that for some classes of bipartite graphs their proper orientation number is less than or equal to 6. We also show that this parameter is at most 4 for graphs which are trees and proof of that fact leads to a polynomial-time algorithm of finding proper orientation of such graphs.
Fiachra Knox, Sebastián González Hermosillo de la Maza, Bojan Mohar, and Cláudia Linhares Sales. Proper Orientations of Planar Bipartite Graphs. pages 2-6, sep 2016. |
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03.11.2016 Krzysztof Francuz, Szymon Łukasz |
Fast and simple connectivity in graph timelines |
Referowana praca (autorstwa J. Łąckiego i A. Karczmarza) rozważa grafy, w którym krawędzie podlegają zmianom - są dodawane bądź usuwane. Opisany jest efektywny algorytm odpowiadający na pytania o osiągalność (istnienie ścieżki między dwoma wierzchołkami) i dwuspójność (istnienie dwóch rozłącznych ścieżek) na zadanych przedziałach czasowych. |
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27.10.2016 Dawid Pyczek, Jakub Rówiński |
Faster deterministic sorting and priority queues in linear space |
Dolne ograniczenie O(n log n) na problem sortowania obowiązuje tylko, jeśli na sortowanych obiektach nie można wykonać żadnych operacji innych niż porównanie. Jeżeli natomiast sortujemy liczby całkowite, możliwe są szybsze algorytmy - referowana praca Mikkela Thorupa z 1997 opisuje algorytm działający w O (n (log log n)2). |
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27.10.2016 Magdalena Gargas |
Optymalizacja Kombinatoryczna The geometry of nesting problems: A tutorial |
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26.10.2016 Wojciech Łopata |
Podstawy Informatyki Universality and Almost Decidability by Cristian S. Calude and Damien Desfontaines |
| We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems. |
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25.10.2016 Marcin Briański |
Kryptologia Unifying Zero-knowledge Proofs of Knowledge |
We present a simple zero-knowledge proof of knowledge protocol of which many protocols in the literature are instantiations. These include Schnorr's protocol for proving knowledge of a discrete logarithm, the Fiat-Shamir and Guillou-Quisquater protocols for proving knowledge of a modular root, protocols for proving knowledge of representations (like Okamoto's protocol), protocols for proving equality of secret values, a protocol for proving the correctness of a Diffie-Hellman key, protocols for proving the multiplicative relation of three commitments (as required in secure multi-party computation), and protocols used in credential systems. This shows that a single simple treatment (and proof), at a high level of abstraction, can replace the individual previous treatments. Moreover, one can devise new instantiations of the protocol. [1] Ueli Maurer, Unifying Zero-knowledge Proofs of Knowledge, Progress in Cryptology – AFRICACRYPT 2009, Vol. 5580 LNCS, pp 272-286
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20.10.2016 Helena Borak |
Optymalizacja Kombinatoryczna Exact algorithms for the two-dimensional strip packing problem with and without rotations |
| We propose exact algorithms for the two-dimensional strip packing problem (2SP) with and without 90 degree rotations. We first focus on the perfect packing problem (PP), which is a special case of 2SP, wherein all given rectangles are required to be packed without wasted space, and design branch-and-bound algorithms introducing several branching rules and bounding operations. A combination of these rules yields an algorithm that is especially efficient for feasible instances of PP. We then propose several methods of applying the PP algorithms to 2SP. Our algorithms succeed in efficiently solving benchmark instances of PP with up to 500 rectangles and those of 2SP with up to 200 rectangles. They are often faster than existing exact algorithms specially tailored for problems without rotations. |
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20.10.2016 Mateusz Twaróg, Patryk Urbański |
Disjoint Set Union with randomized linking |
Algorytm Find-Union w najbardziej znanej wersji implementowany jest przez las zbiorów rozłącznych z kompresją ścieżek i łączeniem według rang. Prezentowana praca, autorstwa Goela, Khanny, Larkina i Tarjana, analizuje złożoność w wersji z arbitralnym (losowym) łączeniem drzew. |
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19.10.2016 Bartosz Walczak |
Informatyka Teoretyczna Common tangents of two disjoint polygons in linear time and constant workspace |
| A tangent of a polygon is a line touching but not crossing the polygon. Two disjoint polygons can have four, two, or no common tangents, depending on whether the convex hulls of the polygons are disjoint, overlapping, or nested. We describe a very simple linear-time constant-workspace algorithm to compute all common tangents of two disjoint polygons, each given by a read-only array of its corners in a cyclic order. This is joint work with Mikkel Abrahamsen. |
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19.10.2016 Pola Kyzioł |
Podstawy Informatyki The domino problem for self-similar structures by Sebastian Barbieri and Mathieu Sablik |
| We defne the domino problem for tilings over self-similar structures of $Z^d$ given by forbidden patterns. In this setting we exhibit non-trivial families of subsets with decidable and undecidable domino problem. |
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18.10.2016 Grzegorz Jurdzinski |
Kryptologia Timing attacks |
Cryptosystems like AES or RSA use algorithms which runtime depends on input data or using CPU cache. Basing on this fact an attacker can find secret keys by choosing inputs and carefully measuring time needed for computations. In this talk I will present such attacks and how to prevent them.
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13.10.2016 Krzysztof Barański |
Optymalizacja Kombinatoryczna Level-Oriented Two-Dimensional Packing Algorithms |
| The paper includes an overview of several algorithms, their complexities and approximation ratios solving two-dimensional strip packing problem: 1) First-Fit Decreasing Height (FFDH) - time complexity: O(nlgn), approximation ratio: <= 17/10 OPT(I) + 1 [with proof] 2) Next-Fit Decreasing Height (NFDH) - time complexity: O(nlgn), approximation ratio: <= 17/10 OPT(I) + 1 [with proof] 3) Best-Fit Decreasing Height (BFDH), Bottom-Left (BL), Steinberg's algorithm, Split-Fit (SF) |
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13.10.2016 Grzegorz Bukowiec, Sylwester Klocek |
Algorytm FKT |
| Rozstrzygnięcie, czy w grafie istnieje skojarzenie, oraz znalezienie takiego skojarzenia są problemami łatwymi obliczeniowo. Liczenie wszystkich skojarzeń jest jednak problemem #P-zupełnym - wielomianowy algorytm na ten problem pociągałby równość P = NP. Istnieje jednak sposób na policzenie skojarzeń dla pewnej klasy grafów - w szczególności, dla wszystkich grafów planarnych. Algorytm taki - zwany od nazwisk twórców algorytmem FKT - wykorzystuje bliski związek między pojęciami (łatwego obliczeniowo) wyznacznika i (trudnego) permanentu macierzy. |
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12.10.2016 Adam Polak |
Informatyka Teoretyczna Why is it hard to beat O(n^2) for Longest Common Weakly Increasing Subsequecnce? |
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11.10.2016 Michał Zieliński |
Kryptologia SafeDeflate: compression without leaking secrets |
| CRIME and BREACH attacks on TLS/SSL leverage the fact that compression ratio is not hidden by encryption to recover content of secrets. We introduce SafeDeflate—a modification of a standard Deflate algorithm which compression ratio does not leak information about secret tokens. The modification is compatible with existing Deflate and gzip decompressors. We introduce a model in which attacker can obtain ciphertexts of arbitrary compressed plaintext containing secret values. Then we prove that SafeDeflate is secure in this model. |
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06.10.2016 Bartłomiej Bosek |
Optymalizacja Kombinatoryczna A new variant of the game of cops and robber |
The talk presents a joint work of Jarosław Grytczuk, Joanna Sokół, Małgorzata Śleszyńska-Nowak. We consider the following metric version of the Cops and Robbers game. Let G be a simple graph and let k≥1 be a fixed integer. In the first round, Cop picks a subset of k vertices B={v1,v2,…,vk} and then Robber picks a vertex u but keeps it in a secret. Then Cop asks Robber for a vector Du(B)=(d1,d2,…,dk) whose components di=dG(u,vi), i=1,2,…,k, are the distances from u to the vertices of B. In the second round, Robber may stay at the vertex u or move to any neighbouring vertex which is kept in a secret. Then Cop picks another k vertices and asks as before for the corresponding distances to the vertex occupied by Robber. And so on in every next round. The game stops when Cop determines exactly the current position of Robber. In that case, she is the winner. Otherwise, Robber is the winner (that is if Cop is not able to localize him in any finite number of rounds). Let ζ(G) denote the least integer k for which Cop has a winning strategy. Notice that this parameter is well defined since the inequality ζ(G)≤|V(G)| holds obviously. |
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05.10.2016 Tomasz Kisielewski |
Podstawy Informatyki Programy które są w stanie przeprowadzać rozumowania o swoich własnościach Proving properties of programs within their language |
Przedstawię wstępną wersję swojego programu badawczego, mającego ====== I will present an initial version of my research program, whosemain goal is to enable proving properties about programs within |
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04.07.2016 |
The 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA'16 Krakow |
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15.06.2016 Piotr Kawałek i Teodor Jerzak |
Podstawy Informatyki Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability by Antoine Genitrini and Cécile Mailler: |
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence. The fundamental breakthrough of this paper is to generalise the previous results for any (reasonable) sequence of integers which enables us, in particular, to solve the above satisfiability question. We also analyse the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new quotient model happens to exhibit a very interesting threshold (or saturation) phenomena.
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09.06.2016 Gwenaël Joret Université Libre de Bruxelles |
Algorytmiczne Aspekty Kombinatoryki Improved Approximation Algorithms for Hitting 3-Vertex Paths |
We study the problem of deleting a minimum cost set of vertices from a |
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08.06.2016 Kamil Pietruszka |
Podstawy Informatyki Regular Combinators for String Transformations by Rajeev Alur Adam Freilich Mukund Raghothaman |
We focus on (partial) functions that map input strings to a monoid such as the set of integers with addition and the set of output strings with concatenation. The notion of regularity for such functions has been defined using two-way finite-state transducers, (one-way) cost register automata, and MSO-definable graph transformations. In this paper, we give an algebraic and machine-independent characterization of this class analogous to the definition of regular languages by regular expressions. When the monoid is commutative, we prove that every regular function can be constructed from constant functions using the combinators of choice, split sum, and iterated sum, that are analogs of union, concatenation, and Kleene *, respectively, but enforce unique (or unambiguous) parsing. Our main result is for the general case of non-commutative monoids, which is of particular interest for capturing regular string-to-string transformations for document processing. We prove that the following additional combinators suffice for constructing all regular functions: (1) the left-additive versions of split sum and iterated sum, which allow transformations such as string reversal; (2) sum of functions, which allows transformations such as copying of strings; and (3) function composition, or alternatively, a new concept of chained sum, which allows output values from adjacent blocks to mix. |
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02.06.2016 http://wms.mat.agh.edu.pl/~knmd/index.php/i-konferencja-naukowa-knmd/harmonogram/ |
Algorytmiczne Aspekty Kombinatoryki Konferencja Studencka na AGH |
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01.06.2016 Szymon Borak |
Informatyka Teoretyczna Polynomial time algorithm for finding Hamiltonian cycles in thin grid graphs |
In general, Hamiltonian Cycle Problem is NP-complete in triangular and square grids. In "Not being(super)thin or solid is hard: A study of grid Hamiltonicity" Arkin et al. claimed HCP for thin triangular grids and thin square grids to be NP-complete as well. However the arguments they gave are incorrect. In fact we show that thin triangular grids as well as thin square grids always have HC. Moreover we show a linear algorithm for finding a HC in such graphs. |
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01.06.2016 Piotr Bejda |
Podstawy Informatyki PATTERN AVOIDANCE IS NOT P RECURSIVE by SCOTT GARRABRANT AND IGOR PAK |
Let F \subset S_k be a finite set of permutations and let C_n (F) denote the number of permutations avoiding the set of patterns F.
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25.05.2016 Kolja Knauer Université Aix-Marseille |
Informatyka Teoretyczna Orienting triangulations - towards Schynyder woods on orientable surfaces |
We show that the edges of any triangulation of a closed surface different from the projective plane and the sphere can be oriented such that every vertex has non-zero outdegree divisble by three. This confirms a conjecture of Barát and Thomassen. We will explain why this is a first step towards the generalization of Schynyder woods from the plane to orientable surfaces and what is know |
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19.05.2016 Miloš Stojaković University of Novi Sad |
Algorytmiczne Aspekty Kombinatoryki Maker-Breaker games on random graphs |
Of all types of positional games, Maker-Breaker games are probably the |
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18.05.2016 Pola Kyzioł |
Podstawy Informatyki NP-Completeness of a Combinator Optimization Problem by M. S. Joy and V. J. Rayward-Smith |
We consider a deterministic rewrite system for combinatory logic over combinators $S,K,I,B,C,S',B'$, and $C'$. |
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28.04.2016 Wojciech Samotij Tel Aviv University |
Algorytmiczne Aspekty Kombinatoryki How does a typical finite metric space look like? |
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27.04.2016 Michał Zieliński |
Podstawy Informatyki Beta Reduction is Invariant, Indeed by Beniamino Accattoli and Ugo Dal Lago |
Slot and van Emde Boas weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time.Is lambda calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda term? This paper presents the first complete positive answer to this longstanding problem. Moreover, our answer is completely machineindependent and based over a standard notion in the theory of lambda calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modelled after linear logic proof nets and admitting a decomposition of leftmostoutermost |
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21.04.2016 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki On some problems in combinatorial number theory |
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20.04.2016 Adam Polak |
Informatyka Teoretyczna On subposets of dimension two |
We study the maximum guaranteed size of a dimension two subposet of an n-element poset. A trivial lower bound of the order of n^{1/2} follows from the Dilworth's theorem. We show an upper bound of the order of n^{2/3} improving the n^{0.8295} result by Reiniger and Yeager. We also discuss promising methods for achieving a better lower bound. |
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20.04.2016 Wojciech Kruk |
Podstawy Informatyki On the equivalence of different presentations of Turner's bracket abstraction algorithm by Lukasz Czajka |
Turner's bracket abstraction algorithm is perhaps the most well-known improvement on simple bracket abstraction algorithms. It is also one of the most studied bracket abstraction algorithms. The definition of the algorithm in Turner's original paper is slightly ambiguous |
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14.04.2016 Michał Farnik Jagiellonian University |
Algorytmiczne Aspekty Kombinatoryki Hat guessing game on sparse graphs |
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13.04.2016 Katarzyna Janocha |
Podstawy Informatyki On the Computing Power of +, -, and x by Marcello Mamino |
Modify the Blum-Shub-Smale model of computation replacing the permitted computational primitives (the real field operations) with any finite set B of real functions semialgebraic over the rationals. Consider the class of Boolean decision problems that can be solved |
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07.04.2016 Steven Chaplick, Universitat Wurzburg |
Algorytmiczne Aspekty Kombinatoryki Intersection Graphs of Non-crossing Paths |
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06.04.2016 Maciej Poleski |
Podstawy Informatyki The Fractal Dimension of SAT Formulas by Carlos Ansotegui, Maria Luisa Bonet, Jesus Giraldez-Cru and Jordi Levy |
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed |
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30.03.2016 Michał Śliwka Teroplan |
Informatyka Teoretyczna Efficient algorithm for several diverse results in public transport routing system |
We present a solution to problem arising in public transport routing systems: |
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30.03.2016 Magdalena Wiercioch |
Podstawy Informatyki Principal types of BCK-lambda-termss by Sachio Hirokawa |
BCK-lambda-terms are the I-terms in which each variable occurs at most once. The principal type of a lambda-term is the most general type of the term. In this paper we prove that if two BCK-lambda-terms in beta-normal form have the same principal type then they are identical. This solves the following problem (Y. Komori, 1987) in more general form: if two closed beta eta-normal form BCK-lambda-terms are assigned to the same minimal BCK-formula, are they identical? A minimal BCK-formula is the most general formula among BCK-provable formulas with respect to substitutions for type variables. To analyze type assignment, the notion of "connection" is introduced. A connection is a series of occurrences of a type. in a type assignment figure. Connected occurrences of a type have the same |
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23.03.2016 Damian Goik |
Informatyka Teoretyczna Direct solver algorithms for systems created on the basis of adaptive meshes |
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23.03.2016 Agnieszka Łupińska |
Podstawy Informatyki PARALLEL STANDARD TRANSLATION BETWEEN LAMBDA CALCULUS AND COMBINATORY LOGIC (wyniki własne) |
The talk is about the parallel approach to the standard translation between Lambda Calculus and Combinatory Logic. Let L be a lambda-term and C be a combinator produced from L by the standard translation. Each lambda abstraction occurring in L, causes the linear expansion of some paths in the tree of the C combinator. We will show that the tree expansion can be performed parallel in logarithmic time on the path length. We will also discuss whether this procedure can be performed in the constant time. |
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17.03.2016 Gabriel Jakóbczak |
Algorytmiczne Aspekty Kombinatoryki Additive chromatic number of several graph families |
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02.02.46210 Jakub Cisło, Grzegorz Jurdziński |
Tight Hardness Results for LCS and other Sequence Similarity Measures |
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03.03.2016 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Pattern avoiding coloring of the plane |
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02.03.2016 Piotr Wójcik |
Podstawy Informatyki Asymptotic properties of first order logic with one binary predicat symbol (wyniki własne) |
Wyniki własne |
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24.02.2016 Zygmunt Łenyk |
Podstawy Informatyki Minimum Propositional Proof Length is NP-Hard to Linearly Approximate (by Michael Alekhnovich, Sam Buss, Shlomo Morany and Toniann Pitassi) |
We prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within a factor of 2^log^{1-o(1)} n. These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs. |
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10.02.2016 William Trotter Georgia Institute of Technology |
Informatyka Teoretyczna Dimension and Cut Vertices |
Motivated by quite recent research involving the relationship between the dimension of a poset and graph theoretic properties of its cover graph, we show that for every $d\ge 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most~$d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. |
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27.01.2016 Michał Dyrek |
Optymalizacja Kombinatoryczna The Linear Arboricity of Graphs |
A linear forest is a forest in which each connected component is a path. The linear arboricity la(G) of a graph G is the minimum number of linear forests whose union is the set of all edges of G. The linear arboricity conjecture asserts that for every simple graph G with maximum degree D, la(G) <= [(D+1)/2]. Although this conjecture received a considerable amount of attention, it has been proven only for D <= 6, D = 8, D = 10 and the best known general upper bound for la(G) is la(G) <= [3D/5] for even D and la(G) <= [(3D + 2)/5] for odd A. Here we prove that for every e > 0 there is a D_0 so that for every G with maximum degree D > D_0, la(G) <= (1/2 + e) * D. To do this, we first prove the conjecture for every G with an even maximum degree D and with girth g > 50*D. N. Alon, The Linear Arboricity of Graphs |
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27.01.2016 Miron Ficak |
Informatyka Teoretyczna On Exact Quantum Query Complexity |
We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms Based on the paper: On Exact Quantum Query Complexity, by Ashley Montanaro, Richard Jozsa and Graeme Mitchison |
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21.01.2016 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Coloring graphs with many colors on cycles |
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20.01.2016 Michał Seweryn |
Informatyka Teoretyczna Data Structures on Event Graphs |
We investigate the behavior of data structures when the input and operations Based on the paper: Data Structures on Event Graphs, by Bernard Chazelle and Wolfgang Mulzer |
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20.01.2016 Pola Kyzioł |
Optymalizacja Kombinatoryczna Matching in regular and almost regular graphs |
I present an O(n^2*log n)-time algorithm that finds a maximum matching in a regular graph with n vertices. More generally, the algorithm runs in O(r*n^2*log n) time if the difference between the maximum degree and the minimum degree is less than r. R. Yuster, Maximum matching in regular and almost regular graphs |
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20.01.2016 Wiktor Tendera |
Podstawy Informatyki Some Remarks on Lengths of Propositional Proofs (by Samuel R. Buss) |
We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depth d Frege proofs of m lines can be transformed into depth d proofs of O(m^{d+1}) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. |
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14.01.2016 Michał Laosń IM PAN, Freie Universitat Berlin |
Algorytmiczne Aspekty Kombinatoryki On the toric ideal of a matroid and related combinatorial problems |
When an ideal is defined only by combinatorial means, one expects to have a combinatorial description of its set of generators. An attempt to achieve this description often leads to surprisingly deep combinatorial questions. White's conjecture is an example. It asserts that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. In the combinatorial language it means that if two multisets of bases of a matroid have equal union (as a multiset), then one can pass between them by a sequence of symmetric exchanges. White's conjecture resisted numerous attempts since its formulation in 1980. We will discuss its relations with other open problems concerning matroids. |
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13.01.2016 Piotr Bejda |
Optymalizacja Kombinatoryczna Perfect matchings in O(n log n) time in regular bipartite graphs |
In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bi-partite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m*sqrt(n)). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to O'(min(m, n^2.5/d)) and then to O'(min(m,n^2/d)). It was also shown that the latter algorithm is optimal up to polylogarithmic factors among all algorithms that use non-adaptive uniform sampling to reduce the size of the graph as a first step. A. Goel and M. Kapralov and S. Khanna, Perfect matchings in O n log n time in regular bipartite graphs |
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13.01.2016 Marcin Zieliński |
Podstawy Informatyki A correspondence between rooted planar maps and normal planar lambda terms (by Noam Zeilberger and Alain Giorgetti) |
A rooted planar map is a connected graph embedded in the plane, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be \beta normal if it is fully reduced, and planar if it uses all of its variables exactly once and in last-in, first-out order. We exhibit a bijection between rooted planar maps and normal planar lambda terms (with one free variable), by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in lambda calculus. |
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07.01.2016 Mateusz Michałek IM PAN, Warszawa, Freie Universitaet, Berlin |
Algorytmiczne Aspekty Kombinatoryki Tensors and algorithms for matrix multiplication |
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16.12.2015 Michał Kosnowski |
Informatyka Teoretyczna Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem |
We examine the problem of determining a spanning tree of a given graph Based on the paper: Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem, by Martin Knauer and Joachim Spoerhase |
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16.12.2015 Krzysztof Barański |
Podstawy Informatyki WORDS IN LINEAR GROUPS, RANDOM WALKS, AUTOMATA AND P-RECURSIVENESS (by SCOTT GARRABRANT AND IGOR PAK) |
Fix a finite set S \suset GL(k, Z). Denote by a_n the number of products of matrices in S of length n that are equal to 1. We show that the sequence a_n is not always P-recursive. This answers a question of Kontsevich. |
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16.12.2015 Krzysztof Kleiner |
Optymalizacja Kombinatoryczna Online Dual Edge Coloring of Paths and Trees |
Extending the results presented on the preceding seminar, we study a dual version of online edge coloring, where the goal is to color as many edges as possible using only a given number, k, of available colors. All of our results are with regard to competitive analysis. For paths, we consider k=2, and for trees, we consider any k>=2. We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms on paths as well as trees. This is the first time that an optimal algorithm for online dual edge coloring has been identified for a class of graphs. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. Again, it is the first time that this has been done for a class of graphs. For trees, we also show that even randomized algorithms cannot be much better than First-Fit. L. M. Favrholdt, J. W. Mikkelsen, Online Dual Edge Coloring of Paths and Trees |
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09.12.2015 Konrad Kalita |
Informatyka Teoretyczna A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows |
A new approach to the minimum-cost integral flow problem for small values of the flow is presented. It reduces the problem to the tests of simple multivariate polynomials over a finite field of characteristic two for non-identity with zero. In effect, we show that a minimum-cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(k log(kn) + log2(kn)) time and using 2k(kn)O(1) processors. Thus, in particular, for the minimum-cost flow of value O(log n), we obtain an RNC2 algorithm, improving upon time complexity of earlier NC and RNC algorithms. Based on the paper: A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows, by Andrzej Lingas and Mia Persson
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09.12.2015 Magdalena Wiercioch |
Podstawy Informatyki A Probabilistic Forest-to-String Model for Language Generation from Typed Lambda Calculus Expressions (by Wei Lu and Hwee Tou Ng) |
This paper describes a novel probabilistic approach for generating natural language sentences from their underlying semantics in the form of typed lambda calculus. The approach is built on top of a novel reduction based weighted synchronous context free grammar formalism, which facilitates the transformation process from typed lambda calculus into natural language sentences. Sentences can then be generated based on such grammar rules with a log linear model. To acquire such grammar rules automatically in an unsupervised manner, we also propose a novel approach with a generative model, which maps from subexpressions of logical forms to word sequences in natural language sentences. Experiments on benchmark datasets for both English and Chinese generation tasks yield significant improvements over results obtained by two state of the art machine translation models, in terms of both automatic metrics and human evaluation. |
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09.12.2015 Mateusz Twaróg |
Optymalizacja Kombinatoryczna On-Line Edge-Coloring with a Fixed Number of Colors |
We investigate a variant of on-line edge-coloring in which there is a fixed number of colors available and the aim is to color as many edges as possible. We prove upper and lower bounds on the performance of different classes of algorithms for the problem. Moreover, we determine the performance of two specific algorithms, First-Fit and Next-Fit. Specifically, algorithms that never reject edges that they are able to color are called fair algorithms. We consider the four combinations of fair/not fair and deterministic/randomized. We show that the competitive ratio of deterministic fair algorithms can vary only between approximately 0.4641 and 1/2 , and that Next-Fit is worst possible among fair algorithms. Moreover, we show that no algorithm is better than 4/7 -competitive. If the graphs are all k-colorable, any fair algorithm is at least 1/2 -competitive. Again, this performance is matched by Next-Fit while the competitive ratio for First-Fit is shown to be k/(2k - 1), which is significantly better, as long as k is not too large. M. Favrholdt, N. Nielsen, On-Line Edge-Coloring with a Fixed Number of Colors, Algorithimca 35 (2), 176-191, 2003 |
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02.12.2015 Grzegorz Bukowiec |
Podstawy Informatyki A \lambda to CL translation for strong normalization (by Yohji AKAMA) |
We introduce a simple translation from \lambda-calculus to combinatory logic (CL) such that: A is an SN \lambda-term iff the translation result of A is an SN term of CL (the reductions are \beta-reduction in \lambda-calculus and weak reduction in CL). None of the conventional translations from \lambda-calculus to CL satisfy the above property. Our translation provides a simpler SN proof of Godel's \lambda-calculus by the ordinal number assignment method. By using our translation, we construct a homomorphism from a conditionally partial combinatory algebra which arises over SN \lambda-terms to a partial combinatory algebra which arises over SN CL-terms. |
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02.12.2015 Helena Borak |
Optymalizacja Kombinatoryczna Linear Extensions of N-free Orders |
We consider the number of linear extensions of an N-free order P. We give upper and lower bounds on this number in terms of parameters of the corresponding arc diagram. We propose a dynamic programming algorithm to calculate the number. The algorithm is polynomial if a new parameter called activity is bounded by a constant. The activity can be bounded in terms of parameters of the arc diagram. Stefan Felsner , Thibault Manneville, Linear Extensions of N-free Orders, Order 32 (2), 147-155, 2015 |
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25.11.2015 Grzegorz Bukowiec |
Podstawy Informatyki A \lambda to CL translation for strong normalization (by Yohji AKAMA) |
We introduce a simple translation from \lambda-calculus to combinatory logic (CL) such that: A is an SN \lambda-term iff the translation result of A is an SN term of CL (the reductions are \beta-reduction in \lambda-calculus and weak reduction in CL). None of the conventional translations from \lambda-calculus to CL satisfy the above property. Our translation provides a simpler SN proof of Godel's \lambda-calculus by the ordinal number assignment method. By using our translation, we construct a homomorphism from a conditionally partial combinatory algebra which arises over SN \lambda-terms to a partial combinatory algebra which arises over SN CL-terms. |
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18.11.2015 Maciej Poleski |
Informatyka Teoretyczna Hitting All Maximal Independent Sets of a Bipartite |
We prove that given a bipartite graph G with vertex set V and an integer k, Based on the paper: Hitting All Maximal Independent Sets of a Bipartite, by Jean Cardinal and Gwenaël Joret |
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18.11.2015 Marcin Kostrzewa |
Podstawy Informatyki Counting a type's prinipal inhabitants (by Broda and Damas) |
We present a Counting Algorithm that computes the number of lambda-terms in \beta-normal form that have a given type as a principal type and produces a list of these terms. The design of the algorithm follows the lines of Ben-Yelles algorithm for counting normal (not neessarily principal) inhabitants of a type. |
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18.11.2015 Leszek Jakub Kania |
Optymalizacja Kombinatoryczna Improved Bounds for Online Preemptive Matching |
When designing a preemptive online algorithm for the maximum matching problem, we wish to maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achievable by preemptive online algorithms. L. Epstein, A. Levin, D. Segev, O. Weimann, Online Preemptive Matching, arXiv 2012 |
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05.11.2015 Adam Gągol Jagiellonian University |
Algorytmiczne Aspekty Kombinatoryki On the Lonely Runner Problem II |
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04.11.2015 Bartłomiej Poleszak |
Podstawy Informatyki Card-Based Protocols for Any Boolean Function (by Takuya Nishida, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone ) |
Card-based protocols that are based on a deck of physical cards achieve secure multi-party computation with information-theoretic secrecy. Using existing AND, XOR, NOT, and copy protocols, one can naively construct a secure computation protocol for any given (multivariable) Boolean function as long as there are plenty of additional cards. However, an explicit sufficient number of cards for computing any function has not been revealed thus far. In this paper, we propose a general approach to constructing an efficient protocol so that six additional cards are sufficient for any function to be securely computed. Further, we prove that two additional cards are sufficient for any symmetric function. |
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04.11.2015 Jakub Cieśla |
Optymalizacja Kombinatoryczna Computing Tree-Depth Faster Than 2^n |
A connected graph has tree-depth at most k if it is a subgraph of the clusure of a rooted tree whose height is at most k. The autors give an algorithm which for a given n-vertex graph G, in time O(1.9602^n) computes the tree-depth of G. The algorithm is based on combinatorial results revealing the structure of minimal rooted trees whose closures contain G. F. V. Fomin, A. C. Giannopoulou, M. Pilipczuk, Computing Tree-Depth Faster Than 2^n, Algorithmica 73 (1), 202-216, 2015 |
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28.10.2015 Ariel Gabizon Technion, Israel |
Informatyka Teoretyczna Quasi-Linear Size Zero Knowledge from Linear-Algebraic PCPs |
A probabilistically checkable proof (PCP) enables, e.g., checking the satisfiability of a 3-SAT formula ɸ, while only examining a constant number of locations in the proof. A long line of research led to the construction of PCPs with length that is quasi-linear in n := |ɸ|. In a zero knowledge PCP with knowledge bound K, reading any K symbols of the proof reveals no additional information besides the validity of the statement; e.g., no information is revealed about the assignment satisfying ɸ. Kilian, Petrank, and Tardos gave a transformation from any PCP into a ZK-PCP with knowledge bound K, for any desired K. A drawback of their transformation is that it requires multiplying the proof length by a factor of (at least) K^6. In this work, we show how to construct PCPs that are zero knowledge for knowledge bound K and of length quasilinear in K and n, provided that the prover is forced to write the proof in two rounds. In this model, which we call duplex PCP (DPCP), the verifier gets an oracle string from the prover, then replies with some randomness, and then gets another oracle string from the prover, and it can make up to K queries to both oracles. Deviating from previous works, our constructions do not invoke the PCP Theorem as a blackbox but rely on algebraic properties of a specific family of PCPs. We show that if the PCP has a certain linear algebraic structure (which many constructions have, including [BFLS91,ALMSS98,BS08]) we can add the zero knowledge property at virtually no cost while introducing only minor modifications in the algorithms of the prover and verifier. We believe that our linear-algebraic characterization of PCPs may be of independent interest, as it gives a simplified way to view previous well-studied PCP constructions. Joint work with Eli Ben-Sasson, Alessandro Chiesa and Madars Virza |
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28.10.2015 Karol Banyś |
Optymalizacja Kombinatoryczna Fast Algorithm for Partial Covers in Words |
In this article autors introduce a new notion of α-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least α positions in w. They develop a data structure of O(n) size (where n=|w|) that can be constructed in O(nlogn) time which they apply to compute all shortest α-partial covers for a given α. They also employ it for an O(nlogn)-time algorithm computing a shortest α-partial cover for each α=1,2,…,n. Tomasz Kociumaka, Solon P. Pissis, Jakub Radoszewski , Wojciech Rytter, Tomasz Waleń, Fast Algorithm for Partial Covers in Words, Algorithmica 73 (1), 217-233, 2015 |
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28.10.2015 Zbigniew Gołębiewski (PWr) |
Podstawy Informatyki On the number of lambda terms with prescribed size of their De Bruijn representation |
John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n) is o( n^−3/2 \tau^−n ) for \tau ≈ 1.963448 . We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is \Theta ( n^−3/2 \rho^−n ) with some class dependent constant \rho, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds |
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21.10.2015 Ariel Gabizon Technion, Israel |
Informatyka Teoretyczna Representative sets for multisets |
In this talk I will explain this notion. Then, to illustrate its usefulness, I will show how it was used by Fomin, Lokshtanov and Saurabh to design a fast algorithm for finding long simple paths in a directed graph. Finally, I will describe a recent work where we generalize the notion of a representative set to a family of multisets and derive algorithmic applications.
Based on the paper Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints with Daniel Lokshtanov and Michał Pilipczuk |
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21.10.2015 Maciej Poleski |
Podstawy Informatyki On the Recursive Enumerability of Fixed-Point Combinators (by Mayer Goldberg) |
We show that the set of fixed-point combinators forms a recursively enumerable subset of a larger set of terms we call non-standard fixedpoint combinators. These terms are observationally equivalent to fixedpoint combinators in any computable context, but the set of non-standard fixed-point combinators is not recursively enumerable. |
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21.10.2015 Paweł Kubiak |
Optymalizacja Kombinatoryczna Lower bounds for dynamic algorithms |
In my presentation I will discus some elementary dynamic problems (Single source reachability and Dynamic diameter) and then I will present interesting reduction from this problems to Orthogonal Vectors Problems. These reductions imply that if it would be possible to solve SSR in O(m^(1-ε)) or do 1.3 approximation of DD in O(m^(2-ε)) then SETH will be refuted. |
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14.10.2015 Katarzyna Janocha |
Optymalizacja Kombinatoryczna Conditional hardness and equivalences for graph problems |
Some graph problems (such as such as APSP, negative triangle, distance product or radius) do not have any known solutions better then the naive ones. We show subquadraic and subcubic reductions between them, proving that in case of finding a faster algorithm for any of the problems would be equivalent of reducing the complexity of each of them. We separate algorithms for sparse and dense graphs and focus on basic methods for both classes. V. Williams, Conditional hardness and equivalences for graph problems |
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14.10.2015 Łukasz Lachowski |
Podstawy Informatyki On the Complexity of the standard translation from Lambda Calculus to Combinatory Logic (wyniki własne) |
Kontynuacja |
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07.10.2015 Zygmunt Łenyk |
Optymalizacja Kombinatoryczna Hardness for Easy Problems (overview) |
Introduction into a young branch of algorithmics. We discuss why we are stuck during developing fast algorithms to some well-known problems. Problems in P and suitable reductions form equivalence classes of problems, inside which improving asymptotic time of any of them would automatically improve the rest. At the bottom of these classes lie problems such as: 3SUM, all-pairs-shortest-paths, orthogonal vectors. Their complexities are guarded by strong conjectures which, if proven wrong, would revoke widely believed conjectures such as SETH. Amir Abboud, Arturs Backurs, Piotr Indyk and Virginia V. Williams, Hardness for easy problems - An introduction, 2015 |
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07.10.2015 Łukasz Lachowski |
Podstawy Informatyki On the Complexity of the standard translation from Lambda Calculus to Combinatory Logic (wyniki własne) |
We investigate the complexity of the standard translation between |
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10.06.2015 Grzegorz Świrski |
Podstawy Informatyki Near semi-rings and lambda calculus by Rick Statman |
| A connection between lambda calculus and the algebra of near semi-rings is discussed. Among the results is the following completeness theorem. A first-order equation in the language of binary associative distributive algebras is true in all such algebras if and only if the interpretations of the first order terms as lambda terms beta-eta convert to one another. A similar result holds for equations containing free variables. |
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03.06.2015 Łukasz Lachowski |
Informatyka Teoretyczna An Algorithmic Characterization of Polynomial Functions over Z_{p^n} |
| In this paper we consider polynomial representability of functions defined over Z_{p^n} , where p is a prime and n is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to determine whether a given function over Z_{p^n} is polynomially representable or not, and (ii) finds the polynomial if it is polynomially representable. The previous characterizations given by Kempner (Trans. Am. Math. Soc. 22(2):240266, 1921) and Carlitz (Acta Arith. 9(1), 6778, 1964) are existential in nature and only lead to an exhaustive search method, i.e. algorithm with complexity exponential in size of the input. Our characterization leads to an algorithm whose running time is linear in size of input. We also extend our result to the multivariate case. References: Ashwin Guha, Ambedkar Dukkipati, An Algorithmic Characterization of Polynomial Functions over Z_{p^n}, Algorithmica (2015) 71:201-218 |
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03.06.2015 Radosław Smyrek |
Podstawy Informatyki Best Response Analysis in Two Person Quantum Games by Azharuddin Shaik, Aden Ahmed |
| In this paper, we find particular use for a maximally entangled initial state that produces a quantized version of two player two strategy games. When applied to a variant of the well-known game of Chicken, our construction shows the existence of new Nash equilibria with the players receiving better payoffs than those found in literature. |
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27.05.2015 Paweł Zegartowski |
Informatyka Teoretyczna Cache-Oblivious Hashing |
| The hash table, especially its external memory version, is one of the most important index structures in large databases. Assuming a truly random hash function, it is known that in a standard external hash table with block size b, searching for a particular key only takes expected average t_q=1+1/2^Ω(b) disk accesses for any load factor α bounded away from 1. However, such near-perfect performance is achieved only when b is known and the hash table is particularly tuned for working with such a blocking. In this paper we study if it is possible to build a cache-oblivious hash table that works well with any blocking. Such a hash table will automatically perform well across all levels of the memory hierarchy and does not need any hardware-specific tuning, an important feature in autonomous databases. We first show that linear probing, a classical collision resolution strategy for hash tables, can be easily made cache-oblivious but it only achieves t_q=1+Θ(α/b) even if a truly random hash function is used. Then we demonstrate that the block probing algorithm (Pagh et al. in SIAM Rev. 53(3):547558, 2011) achieves t_q=1+1/2^Ω(b), thus matching the cache-aware bound, if the following two conditions hold: (a) b is a power of 2; and (b) every block starts at a memory address divisible by b. Note that the two conditions hold on a real machine, although they are not stated in the cacheoblivious model. Interestingly, we also show that neither condition is dispensable: if either of them is removed, the best obtainable bound is t_q=1+O(α/b), which is exactly what linear probing achieves. References:Rasmus Pagh, ZheweiWei, Ke Yi, Qin Zhang, Cache-Oblivious Hashing, Algorithmica (2014) 69:864-883 |
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27.05.2015 Bartłomiej Ryniec |
Podstawy Informatyki GENERIC COMPLEXITY OF UNDECIDABLE PROBLEMS by ALEXEI G. MYASNIKOV AND ALEXANDER N. RYBALOV |
| In this paper we study generic complexity of undecidable problems. It turns out that some classical undecidable problems are, in fact, strongly undecidable, i.e., they are undecidable on every strongly generic subset of inputs. For instance, the classical Halting Problem is strongly undecidable. Moreover, we prove an analog of the Rice's theorem for strongly undecidable problems, which provides plenty of examples of strongly undecidable problems. Then we show that there are natural super-undecidable problems, i.e., problem which are undecidable on every generic (not only strongly generic) subset of inputs. In particular, there are finitely presented semigroups with super-undecidable word problem. To construct strongly- and super-undecidable problems we introduce a method of generic amplification (an analog of the amplification in complexity theory). Finally, we construct absolutely undecidable problems, which stay undecidable on every non-negligible set of inputs. Their construction rests on generic immune sets. |
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20.05.2015 Łukasz Majcher |
Informatyka Teoretyczna List Coloring in the Absence of a Linear Forest |
| The k-COLORING problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The LIST k-COLORING problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,...,k}. Let P_n denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that LIST k-COLORING can be solved in polynomial time for graphs with no induced rP_1+P_5, hereby extending the result of Hoàng, Kami´nski, Lozin, Sawada and Shu for graphs with no induced P_5. Our result is tight; we prove that for any graph H that is a supergraph of P_1 + P_5 with at least 5 edges, already LIST 5-COLORING is NP-complete for graphs with no induced H. References:Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, List Coloring in the Absence of a Linear Forest, Algorithmica (2015) 71:2135 |
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13.05.2015 Krzysztof Kulig |
Informatyka Teoretyczna Metrical Service Systems with Multiple Servers |
| The problem of metrical service systems with multiple servers ((k,l)-MSSMS), proposed by Feuerstein (LATIN'98: Theoretical Informatics, Third Latin American Symposium, 1998), is to service requests, each of which is an l-point subset of a metric space, using k servers in an online manner, minimizing the distance traveled by the servers. We prove that Feuerstein's deterministic algorithm for (k,l)- MSSMS actually achieves an improved competitive ratio of k\cdot({k+l}\choose{l})-1) on uniform metrics. References:Ashish Chiplunkar, Sundar Vishwanathan, Metrical Service Systems with Multiple Servers, Algorithmica (2015) 71:219231 |
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13.05.2015 Bartosz Badura |
Podstawy Informatyki Havannah and TwixT are pspace-complete by Édouard Bonnet, Florian Jamain, and Abdallah Saffidine |
| Numerous popular abstract strategy games ranging from hex and havannah to lines of action belong to the class of connection games. Still, very few complexity results on such games have been obtained since hex was proved pspace-complete in the early eighties. We study the complexity of two connection games among the most widely played. Namely, we prove that havannah and twixt are pspacecomplete. The proof for havannah involves a reduction from generalized geography and is based solely on ring-threats to represent the input graph. On the other hand, the reduction for twixt builds up on previous work as it is a straightforward encoding of hex. |
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06.05.2015 Maciej Solon |
Informatyka Teoretyczna Minimum Fill-in of Sparse Graphs: Kernelization and Approximation |
| The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k^{3/2}) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log k) on planar graphs and O(√k·log k) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for MINIMUM FILL-IN on sparse graphs. References:Fedor V. Fomin, Geevarghese Philip, Yngve Villanger; Minimum Fill-in of Sparse Graphs: Kernelization and Approximation, Algorithmica (2015) 71:120 |
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06.05.2015 Leszek Jakub Kania |
Podstawy Informatyki Fast algorithm finding the shortest reset words by A. Kisielewicz J. Kowalski, and M. Szykuła |
| In this paper we present a new fast algorithm finding minimal reset words for finite synchronizing automata. The problem is know to be computationally hard, and our algorithm is exponential. Yet, it is faster than the algorithms used so far and it works well in practice. The main idea is to use a bidirectional BFS and radix (Patricia) tries to store and compare resulted subsets. We give both theoretical and practical arguments showing that the branching factor is reduced efficiently. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with n states and 2 input letters. We follow Skvorsov and Tipikin, who have performed such a study using a SAT solver and considering automata up to n = 100 states. With our algorithm we are able to consider much larger sample of automata with up to n = 300 states. In particular, we obtain a new more precise estimation of the expected length of the shortest reset word = 2.5 sqrt{n − 5}. |
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29.04.2015 Agnieszka Łupińska |
Informatyka Teoretyczna Strong Conflict-Free Coloring for Intervals |
| We consider the k-strong conflict-free (k-SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval I of the family there are at least k colors each appearing exactly once in I .We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when k=1 and 5−2/k when k≥2. In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any k≥1. We also provide, in case k=O(polylog(n)), a quasipolynomial time algorithm to decide the existence of a k-SCF coloring that uses at most q colors. References:Panagiotis Cheilaris, Luisa Gargano, Adele A. Rescigno, Shakhar Smorodinsky, Strong Conflict-Free Coloring for Intervals, Algorithmica (2014) 70:732-749 |
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29.04.2015 Marcin Kostrzewa |
Podstawy Informatyki A Short Note on Type-inhabitation: Formula-Trees vs. Game Semantics by S. Alves, S. Broda |
| This short note compares two different methods for exploring type-inhabitation in the simply typed lambda-calculus, highlighting their similarities. |
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22.04.2015 Marcin Regdos |
Informatyka Teoretyczna An O(n^4) Time Algorithm to Compute the Bisection Width of Solid Grid Graphs |
| The bisection problem asks for a partition of the n vertices of a graph into two sets of size at most \ceil{n/2}, so that the number of edges connecting the sets is minimised. A grid graph is a finite connected subgraph of the infinite two-dimensional grid. It is called solid if it has no holes. Papadimitriou and Sideri (Theory Comput Syst 29:97110, 1996) gave an O(n^5) time algorithm to solve the bisection problem on solid grid graphs. We propose a novel approach that exploits structural properties of optimal cuts within a dynamic program. We show that our new technique leads to an O(n^4) time algorithm. References:Andreas Emil Feldmann, Peter Widmayer, An O(n^4) Time Algorithm to Compute the Bisection Width of Solid Grid Graphs, Algorithmica (2015) 71:181-200 |
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22.04.2015 Agnieszka Łupińska |
Podstawy Informatyki The Converse principal Type Algorithm by Roger Hindley |
| One chapter from the book Basic Simple Type Theory |
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15.04.2015 Maciej Bendkowski |
Informatyka Teoretyczna Contention Resolution under Selfishness |
| In many communications settings, such as wired and wireless local-area networks, when multiple users attempt to access a communication channel at the same time, a conflict results and none of the communications are successful. Contention resolution is the study of distributed transmission and retransmission protocols designed to maximize notions of utility such as channel utilization in the face of blocking communications. An additional issue to be considered in the design of such protocols is that selfish users may have incentive to deviate from the prescribed behavior, if another transmission strategy increases their utility. The work of Fiat et al. (in SODA'07, pp.179188, SIAM, Philadelphia 2007) addresses this issue by constructing an asymptotically optimal incentive-compatible protocol. However, their protocol assumes the cost of any single transmission is zero, and the protocol completely collapses under non-zero transmission costs. In this paper we treat the case of non-zero transmission cost c.We present asymptotically optimal contention resolution protocols that are robust to selfish users, in two different channel feedback models. Our main result is in the Collision Multiplicity Feedback model, where after each time slot, the number of attempted transmissions is returned as feedback to the users. In this setting, we give a protocol that has expected cost Θ(n+c·log n) and is in o(1)-equilibrium, where n is the number of users. References:George Christodoulou, Katrina Ligett, Evangelia Pyrga, Contention Resolution under Selfishness, Algorithmica (2014) 70:675693 |
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15.04.2015 Agnieszka Łupińska |
Podstawy Informatyki The principal Type Algorithm by Roger Hindley |
| One chapter from the book Basic Simple Type Theory |
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14.04.2015 Maciej Solon |
Algorytmy Randomizowane i Aproksymacyjne Graphs defined by forbidden patterns. |
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01.04.2015 Maciej Bendkowski |
Podstawy Informatyki Über Tautologien, in welchen keine Variable mehr als zweimal vorkommt von S. Jaśkowski |
| CONTINUATION |
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25.03.2015 Lech Duraj, Grzegorz Gutowski, Jakub Kozik |
Informatyka Teoretyczna Chip games and paintability |
| We present a natural family of chip games with strong ties to paintability, on-line 2-coloring of hypergraphs and Maker-Braker games. We solve some of those games and as a result we obtain interesting results in aforementioned areas. One of those results is that the difference between paintabilty and choosability of a graph can be arbitrarily large. |
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18.03.2015 Jarosław Duda |
Informatyka Teoretyczna Asymmetric Numeral Systems: adding fractional bits to Huffman coder |
| Entropy coding is an integral part of most data compression systems. There were previously used mainly two approaches: Huffman coding which is fast but approximates probabilities with powers of 1/2 (suboptimal compression ratio), and arithmetic coding which uses nearly accurate probabilities at cost of being an order of magnitude slower (more expensive). I will talk about new approach: Asymmetric Numeral Systems (ANS), which while using nearly accurate probabilities, has turned out to allow for even faster implementations than Huffman coding. Consequently, succeeding compressors have already switched to ANS in recent months. |
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18.03.2015 Agnieszka Łupińska |
Podstawy Informatyki The principal Type Algorithm by Roger Hindley |
| One chapter from the book Basic Simple Type Theory |
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11.03.2015 Piotr Danilewski Universität des Saarlandes |
Informatyka Teoretyczna AnyDSL - a host for any language |
| In a multi-domain project, there is no single programming language that can be used to program everything. Instead, a combination of general-purpose and domain-specific languages (DSLs) are used. Unfortunately, many domains lack a good, representative DSL, due to domain diversity and difficulty of creating a new compiler. Moreover, the communication across the languages is limited, often requiring the data to be serialized, and reducing the quality of optimization and compile-time verification. In our talk we present our approach to solve these problems, by introducing a new metamorphic language - AnyDSL. The parsing and execution of AnyDSL can be interleaved and the latter can influence the former - e.g. by introducing a new grammar with which parsing should resume. Regardless of the language the source is written in, all code is translated into a low-level, functional representation in continuous passing style (AIR). AIR serves as a "meeting point", permitting inter-DSL communication. AIR can be interpreted or compiled to LLVM bytecode and then to machine code. New grammars are defined also within AnyDSL. Unlike typical parser generators, grammar productions and actions are given as functions. AIR supports dynamic staging - a flexible way to define partial evaluation strategies. With it the overhead of having multiple layers of languages can be resolved early. It also allows the DSL designer to specify domain specific optimizations. After all those transformations, AIR can be compiled to machine code that is efficient, with performance comparable to programs written in general-purpose languages. In our talk we present a new metamorphic language - AnyDSL. AnyDSL permits the native parser to be exchanged with a custom DSL. Regardless of the DSL however, all code is translated into a low-level, functional representation in continuous passing style (AIR). New grammars are defined also within AnyDSL, but unlike typical parser generators, grammar productions and actions are given as functions. AIR supports dynamic staging - a flexible way to define partial evaluation strategies to resolve overheads and define domain specific optimizations. AIR can be compiled to machine code, and produced programs have performance comparable to those produced by general-purpose languages. |
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04.03.2015 Maciej Bendkowski |
Podstawy Informatyki Über Tautologien, in welchen keine Variable mehr als zweimal vorkommt von S. Jaśkowski |
| H. Thiele hat im Jahre 1960 das Problem gestellt, das implikative Teilsystem des Aussagenkalküls mit dem Axiomen B,C,K zu untersuchen. Hier wird für dieses System und für ein anderes, in dem das letzte Axiom durch ein schwächeres, nämlich I ersetzt wird, ein Entscheidungsverfahren angegeben. Die Methode beruht auf einer Untersuchung von gewissen allgemeinen Eigenschaften der Ausdrücke, in welchen keine Satzvariable mehr als zweimal vorkommt. Dabei wird eine dreiwertige Matrix benutzt. |
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28.01.2015 Michał Zając |
Informatyka Teoretyczna Improved Explicit Data Structures in the Bitprobe Model |
| Buhrman et al. [SICOMP 2002] studied the membership problem in the bitprobe model, presenting both randomized and deterministic schemes for storing a set of size n from a universe of size m such that membership queries on the set can be answered using t bit probes. Since then, there have been several papers focusing on deterministic schemes, especially for the first non-trivial case when n=2. The most recent, due to Radhakrishnan, Shah, and Shannigrahi [ESA 2010], describes non-explicit schemes (existential results) for t≥3 using probabilistic arguments. We describe a fully explicit scheme for n=2 that matches their space bound of Θ(m^{2/5}) bits for t=3 and, furthermore, improves upon it for t>3, answering their open problem. Our structure (consisting of query and storage algorithms) manipulates blocks of bits of the query element in a novel way that may be of independent interest. We also describe recursive schemes for n≥3 that improve upon all previous fully explicit schemes for a wide range of parameters. References:Moshe Lewenstein, J. Ian Munro, Patrick K. Nicholson and Venkatesh Raman, Improved Explicit Data Structures in the Bitprobe Model, ESA 2014, LNCS 8737, pp. 630–641, 2014 |
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28.01.2015 21.01.2015,Radosław Smyrek |
Podstawy Informatyki Symmetry groups of boolean functions by Mariusz Grech, Andrzej Kisielewicz |
| We prove that every abelian permutation group, but known exceptions, is the symmetry group of a boolean function. This solves the problem posed in the book by Clote and Kranakis. In fact, our result is proved for a larger class of permutation groups, namely, for all subgroups of direct sums of regular permutation groups. |
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22.01.2015 Adam Polak |
Algorytmiczne Aspekty Kombinatoryki Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees |
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21.01.2015 Bartosz Badura |
Kryptologia A Formal Treatment of Onion Routing |
| Anonymous channels are necessary for a multitude of privacy-protecting protocols. Onion routing is probably the best known way to achieve anonymity in practice. However, the cryptographic aspects of onion routing have not been sufficiently explored: no satisfactory definitions of security have been given, and existing constructions have only had ad-hoc security analysis for the most part. We provide a formal definition of onion-routing in the universally composable framework, and also discover a simpler definition (similar to CCA2 security for encryption) that implies security in the UC framework. We then exhibit an efficient and easy to implement construction of an onion routing scheme satisfying this definition. References:J. Camenisch, A. Lysyanskaya, A Formal Treatment of Onion Routing, Proc CRYPTO'05, pp. 169--187 |
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21.01.2015 Andrzej Głuszyński |
Informatyka Teoretyczna Data Structures for Storing Small Sets in the Bitprobe Model |
| We study the following set membership problem in the bit probe model: given a set S from a finite universe U, represent it in memory so that membership queries of the form "Is x in S?" can be answered with a small number of bitprobes. We obtain explicit schemes that come close to the information theoretic lower bound of Buhrman et al. [STOC 2000, SICOMP 2002] and improve the results of Radhakrishnan et al. [ESA 2001] when the size of sets and the number of probes is small. We show that any scheme that stores sets of size two from a universe of size m and answers membership queries using two bitprobes requires space Ω(m^{4/7}). The previous best lower bound (shown by Buhrman et al. using information theoretic arguments) was Ω(√m). The same lower bound applies for larger sets using standard padding arguments. This is the first instance where the information theoretic lower bound is found to be not tight for adaptive schemes. We show that any non-adaptive three probe scheme for storing sets of size two from a universe of size m requires Ω(√m) bits of memory. This extends a result of Alon and Feige [SODA 2009] to small sets. References:Jaikumar Radhakrishnan, Smit Shah and Saswata Shannigrahi, Data Structures for Storing Small Sets in the Bitprobe Model, ESA 2010, Part II, LNCS 6347, pp. 159–170, 2010. |
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20.01.2015 Maciej Poleski |
Optymalizacja Kombinatoryczna An online version of Rota's basis conjecture |
| Rota's basis conjecture states that in any square array of vectors whose rows are bases of a fixed vector space the vectors can be rearranged within their rows in such a way that afterwards not only the rows are bases, but also the columns. We discuss an online version of this conjecture, in which the permutation used for rearranging the vectors in a given row must be determined without knowledge of the vectors further down the array. The paper contains surprises both for those who believe this online basis conjecture at first glance, and for those who disbelieve it. References:Guus P. Bollen, Jan Draisma, An online version of Rota's basis conjecture, Journal of Algebraic Combinatorics, October 2014 |
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14.01.2015 Konrad Witaszczyk |
Kryptologia How to Re-initialize a Hash Chain |
| Hash Chains are used extensively in various cryptographic systems such as one-time passwords, server supported signatures, secure address resolution, certificate revocation, micropayments etc. However, currently they suffer from the limitation that they have a finite number of links which when exhausted requires the system to be re-initialized. In this paper, we present a new kind of hash chain which we call a Re-initializable Hash Chain (RHC). A RHC has the property that if its links are exhausted, it can be securely re-initialized in a non-repudiable manner to result in another RHC. This process can be continued indefinitely to give rise to an infinite length hash chain, or more precisely, an infinite number of finite length hash chains tied together. Finally we illustrate how a conventional hash chain (CHC) may be profitable replaced with a RHC in cryptographic systems. References:Leslie Lamport, Password Authentication with Insecure Communication, PDF Yuanchao Zhao, Daoben Li, An Improved Elegant Method to Re-initialize Hash Chains, PDF Vipul Goyal, How to Re-initialize a Hash Chain, PDF |
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14.01.2015 Andrzej Dorobisz |
Informatyka Teoretyczna Scheduling parallel jobs to minimize the makespan |
| We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time. References:Johannes Berit, Scheduling parallel jobs to minimize the makespan, J of Schedulling, 9(2006), 433–452 |
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14.01.2015 Bartłomiej Ryniec |
Podstawy Informatyki Infinite time Turing machines with only one tape by Joel David Hamkins, Daniel Evan Seabold |
| Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions f:R → N, the same class of computable functions. Nevertheless, there are infinite time computable functions f:R→R that are not one-tape computable, and so the two models of infinitary computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition yields the full class of all infinite time computable functions. Finally, every ordinal which is clockable by an infinite time Turing machine is clockable by a one-tape machine, except certain isolated ordinals that end gaps in the clockable ordinale |
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13.01.2015 Helena Borak |
Optymalizacja Kombinatoryczna Variants of Hat Guessing Games |
| Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the hat they are wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat-guessing problems with a finite number of players. References:S.Butler, M.T.Hajiaghayi, R.D.Kleinberg, T.Leighton, Hat Guessing Games |
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13.01.2015 Andrzej Dorobisz. |
Algorytmy Randomizowane i Aproksymacyjne Treewidth. Courcelle's theorem. |
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07.01.2015 Paweł Zegartowski |
Kryptologia The Padding Oracle attacks: theoretical background with practical exemplification |
| In many standards, such as. SSL/TLS, IPSEC, WTLS, messages are first pre-formatted, then encrypted in CBC mode with a block cipher. Decryption needs to check if the format is valid. Validity of the format is easily leaked from communication protocols in a chosen ciphertext attack since the receiver usually sends an acknowledgment or an error message.This is a side channel. Since year 2002 the padding oracle attacks are known to be a working example of Chosen Ciphertext Attack possible to perform on various real-world cryptosystems using padding in their vital areas of calculation. The lecture attempts to describe the nature of a padding oracle attack and to point drawbacks of cryptosystems that make them vulnerable for attack of this kind. Moreover the POODLE attack shall be presented as an example of practical application of padding oracle attack against the SSLv3 protocol possible to be used also against servers using newer security protocols (like TLS 1.x). References:Serge Vaudenay, Security Flaws Induced by CBC Padding Applications to SSL, IPSEC, WTLS, EUROCRYPT 2002. Juliano Rizzo, Thai Duong, Practical Padding Oracle Attacks, USENIX WOOT 2010 Möller, Bodo; Duong, Thai; Kotowicz, Krzysztof, This POODLE Bites: Exploiting The SSL 3.0 Fallback, Google Security Advisory 2014 |
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07.01.2015 Łukasz Kapica |
Informatyka Teoretyczna On an on-line scheduling problem for parallel jobs |
| The non-preemptive on-line scheduling of parallel jobs is addressed. In particular we assume that the release dates and the processing times of the jobs are unknown. It is already known that for this problem Garey and Graham's list scheduling algorithm achieves the competitive factor 2−1/m for the makespan if m identical machines are available and if each job requires only a single machine for processing. Here, we show that the same factor also holds in the case of parallel jobs. References:Edwin Naroska, Uwe Schwiegelshohn, On an on-line scheduling problem for parallel jobs, Information Processing Letters, 81(2002), 297–304. |
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07.01.2015 Michał Seweryn |
Podstawy Informatyki A Formalisation of the Myhill-Nerode Theorem Based on Regular Expressions by Chunhan Wu, Xingyuan Zhang, Christian Urban |
| There are numerous textbooks on regular languages. Many of them focus on finite automata for proving properties. Unfortunately, automata are not so straightforward to formalise in theorem provers. The reason is that natural representations for automata are graphs, matrices or functions, none of which are inductive datatypes. Regular expressions can be defined straightforwardly as a datatype and a corresponding reasoning infrastructure comes for free in theorem provers. We show in this paper that a central result from formal language theory—the Myhill-Nerode Theorem—can be recreated using only regular expressions. From this theorem many closure properties of regular languages follow. |
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17.12.2014 Łukasz Majcher |
Kryptologia Searching for Elements in Black Box Fields and Applications |
| We introduce the notion of a black box field and discuss the problem of explicitly exposing field elements given in a black box form. We present several sub-exponential algorithms for this problem using a technique due to Maurer. These algorithms make use of elliptic curves over finite fields in a crucial way. We present three applications for our results: (1) We show that any algebraically homomorphic encryption scheme can be broken in expected sub-exponential time. The existence of such schemes has been open for a number of years. (2) We give an expected sub-exponential time reduction from the problem of finding roots of polynomials over finite fields with low straight line complexity (e.g. sparse polynomials) to the problem of testing whether such polynomials have a root in the field. (3) We show that the hardness of computing discrete-log over elliptic curves implies the security of the Diffie-Hellman protocol over elliptic curves. Finally in the last section of the paper we prove the hardness of exposing black box field elements in a field of characteristic zero. References:Dan Boneh, Richard J. Lipton, Algorithms for Black-Box Fields and their Application to Cryptography, Proceeding CRYPTO'96 pp. 283--297 |
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17.12.2014 Bartosz Wlaczak |
Informatyka Teoretyczna Minors and dimension |
| Streib and Trotter proved in 2012 that posets with bounded height and with planar cover graphs have bounded dimension. Later, Joret et al. proved that the dimension is bounded for posets with bounded height whose cover graphs have bounded tree-width. Recently, I proved that posets of bounded height whose cover graphs exclude a fixed (topological) minor have bounded dimension. This generalizes both the aforementioned results and verifies a conjecture of Joret et al. In this talk, I will introduce the problems of bounding the dimension of posets with sparse cover graphs and the main structural theorems used in the proof of the latter result: the Robertson-Seymour and Grohe-Marx structural decomposition theorems. I will also briefly describe the idea of the proof. |
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17.12.2014 Agnieszka Łupińska |
Podstawy Informatyki Relevant Logic and the Philosophy of Mathematics by Edwin Mares |
| This paper sets out three programmes that attempt to use relevant logic as the basis for a philosophy of mathematics. Although these three programmes do not exhaust the possible approaches to mathematics through relevant logic, they are fairly representative of the current state of the field. The three programmes are compared and their relative strengths and weaknesses set out. At the end of the paper I examine the consequences of adopting each programme for the realist debate about mathematical objects. |
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16.12.2014 Marcin Dziaduś |
Optymalizacja Kombinatoryczna Five-list-coloring of planar graphs |
| Let G be a plane graph with outer cycle C, let u,v be vertices of C and let (L(x):x in V(G)) be a family of sets such that |L(u)|=|L(v)|=2, L(x) has at least three elements for every vertex x of C \ {u,v} and L(x) has at least five elements for every vertex x of G \ V(C). We prove a conjecture of Hutchinson that G has a proper coloring f such that f(x) belongs to L(x) for every vertex x of G. References:Luke Postle, Robin Thomas, Five-list-coloring graphs on surfaces I. Two lists of size two in planar graphs, Journal of Combinatorial Theory, Series B |
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16.12.2014 Grzegorz Gutowski. |
Algorytmy Randomizowane i Aproksymacyjne s-t orientations of planar graphs. |
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10.12.2014 Krzysztof Kulig |
Kryptologia How to Leak a Secret |
| In this paper we formalize the notion of a ring signature, which makes it possible to specify a set of possible signers without revealing which member actually produced the signature.Unlike group signatures, ring signatures have no group managers, no setup procedures, no revocation procedures, and no coordination:any user can choose any set of possible signers that includes himself,and sign any message by using his secret key and the others' public keys,without getting their approval or assistance. Ring signatures provide an elegant way to leak authoritative secrets in an anonymous way, to sign casual email in a way which can only be verified by its intended recipient, and to solve other problems in multiparty computations. The main contribution of this paper is a new construction of such signatures which is unconditionally signer-ambiguous, provably secure in the random oracle model,and exceptionally efficient:adding each ring member increases the cost of signing or verifying by a single modular multiplication and a single symmetric encryption. References:Ronald L. Rivest, Adi Shamir, Yael Tauman, How to Leak a Secret, Advances in Cryptology — ASIACRYPT 2001 LNCS vol. 2248, 2001, pp 552-565 |
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10.12.2014 26.11.2014,Tomasz Kołodziejski |
Informatyka Teoretyczna Opaque sets or how to find a pipe |
| We'll tackle the problem of finding the smallest set in a given class that meets every line intersecting a given convex set. Such a set is know as a barrier. Particularly interesting barrier classes are: connected sets, poly-lines and arbitrary segment barriers. The algorithmic approach yields various approximation constants around 1.6. Little is known about the exact barriers even for simple figures. Algorithms and proofs will be presented most of which require only basic planar geometry knowledge will little calculus (Cauchy surface area formula will be presented with no proof). |
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10.12.2014 Pierre Lescanne (l'École Normale Supérieure de Lyon) |
Podstawy Informatyki Boltzmann samplers |
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09.12.2014 Karol Banyś |
Optymalizacja Kombinatoryczna Online Load Balancing and Correlated Randomness |
| This paper looks at online load balancing, in a setting where each job can only be served by a subset of the servers. The subsets are revealed only on arrival, and can be arbitrary. The cost of an allocation is the sum of cost for each server, which in turn is a convex increasing function of the number of jobs allocated to it. There are no departures. References:S. Moharir, S. Sanghavi. Online Load Balancing and Correlated Randomness. 50th Annual Allerton Conference, 2012 U. Vazirani V. Vazirani A. Mehta, A. Saberi. Adwords and generalized on-line matching. Proceedings of FOCS, 2005 |
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04.12.2014 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Problems and results in combinatorial number theory |
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03.12.2014 Piotr Bejda |
Kryptologia Using hash functions as a hedge against chosen ciphertext attack |
| The cryptosystem recently proposed by Cramer and Shoup is a practical public key cryptosystem that is secure against adaptive chosen ciphertext attack provided the Decisional Diffie-Hellman assumption is true. Although this is a reasonable intractability assumption, it would be preferable to base a security proof on a weaker assumption, such as the Computational Diffie-Hellman assumption. Indeed, this cryptosystem in its most basic form is in fact insecure if the Decisional Diffie-Hellman assumption is false. In this paper we present a practical hybrid scheme that is just as efficient as the scheme of of Cramer and Shoup; indeed, the scheme is slightly more efficient than the one originally presented by Cramer and Shoup; we prove that the scheme is secure if the Decisional Diffie-Hellman assumption is true; we give strong evidence that the scheme is secure if the weaker, Computational Diffie-Hellman assumption is true by providing a proof of security in the random oracle model. References:R. Cramer and V. Shoup. A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In Advances in Cryptology - Crypto'98, pages 13–25, 1998 V. Shoup. Using hash functions as a hedge against chosen ciphertext attack, in Proc. Eurocrypt 2000 |
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03.12.2014 Agnieszka Łupińska |
Podstawy Informatyki General information in relevant logic by Edwin D. Mares |
| There are numerous textbooks on regular languages. Many of them focus on finite automata for proving properties. Unfortunately, automata are not so straightforward to formalise in theorem provers. The reason is that natural representations for automata are graphs, matrices or functions, none of which are inductive datatypes. Regular expressions can be defined straightforwardly as a datatype and a corresponding reasoning infrastructure comes for free in theorem provers. We show in this paper that a central result from formal language theory—the Myhill-Nerode Theorem—can be recreated using only regular expressions. From this theorem many closure properties of regular languages follow. |
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02.12.2014 Andrzej Dorobisz |
Optymalizacja Kombinatoryczna Random Walks that Find Perfect Objects and the Lov´asz Local Lemma |
| We give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly. Our work is inspired by Moser's entropic method proof of the Lov´asz Local Lemma (LLL) for satisfiability and completely bypasses the Probabilistic Method formulation of the LLL. In particular, our method works when the underlying state space is entirely unstructured. Similarly to Moser's argument, the key point is that algorithmic progress is measured in terms of entropy rather than energy (number of violated constraints) so that termination can be established even under the proliferation of states in which every step of the algorithm (random walk) increases the total number of violated constraints. References:Dimitris Achlioptas, Fotis Iliopoulos, Random Walks that Find Perfect Objects and the Lovasz Local Lemma, FOCS 2014 |
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27.11.2014 Grzegorz Guśpiel |
Algorytmiczne Aspekty Kombinatoryki Homomorphisms of Edge-coloured Graphs |
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26.11.2014 Pola Kyzioł |
Kryptologia Another look at non-standard discrete log and Diffie-Hellman problems |
| We examine several versions of the one-more-discrete-log and one-more-Diffie-Hellman problems. In attempting to evaluate their intractability, we find conflicting evidence of the relative hardness of the different problems. Much of this evidence comes from natural families of groups associated with curves of genus 2, 3, 4, 5, and 6. This leads to questions about how to interpret reductionist security arguments that rely on these non-standard problems. References:N. Koblitz, A. Menezes, Another look at non-standard discrete log and Diffie-Hellman problems, J. Math. Cryptology 2 (2008), pp. 311--326 |
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26.11.2014 Konrad Witaszczyk |
Podstawy Informatyki Problems of Proof compexity by Jan Krajicek, Stephen A. Cook and Robert A. Reckhow |
| The ultimate goal of proof complexity is to show that there is no universal propositional proof system allowing for efficient proofs of all tautologies. This is equivalent to showing that the computational complexity class NP is not closed under the complementation. By the universality propositional proof systems subsume methods from other parts of mathematics used for proving the non-existence statements. Because of this, even the partial results known at present (lower bounds for some specific proof systems) revealed interesting links of proof complexity to logic, algebra, combinatorics, computational complexity. We will explain some basic points of proof complexity and give few informal examples in order to motivate the main concepts and problems of proof complexity. |
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25.11.2014 18.11.2014,Jakub Brzeski |
Optymalizacja Kombinatoryczna Markov Chains and Random Walks on Graphs |
| References:D. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph, 2014. L. Lovász, Random walks on graphs: a survey, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), 353–397, Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996. |
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25.11.2014 18.11.2014,Patryk Mikos |
Algorytmy Randomizowane i Aproksymacyjne Kernelization and Linear Programming Techniques |
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19.11.2014 Patryk Mikos |
Kryptologia A practical public key cryptosystem probably secure against adaptive chosen ciphertext attack |
| A new public key cryptosystem is proposed and analyzed. The scheme is quite practical, and is provably secure against adaptive chosen ciphertext attack under standard intractability assumptions. There appears to be no previous cryptosystem in the literature that enjoys both of these properties simultaneously. |
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19.11.2014 Bartosz Badura |
Podstawy Informatyki On the Complexity of Trick-Taking Card Games by Edouard Bonnet, Florian Jamain, and Abdallah Saffidine |
| Determining the complexity of perfect information trick-taking card games is a long standing open problem. This question is worth addressing not only because of the popularity of these games among human players, e.g., DOUBLE DUMMY BRIDGE, but also because of its practical importance as a building block in state-of-the-art playing engines for CONTRACT BRIDGE, SKAT, HEARTS, and SPADES. We define a general class of perfect information twoplayer trick-taking card games dealing with arbitrary numbers of hands, suits, and suit lengths. We investigate the complexity of determining the winner in various fragments of this game class. Our main result is a proof of PSPACE-completeness for a fragment with bounded number of hands, through a reduction from Generalized Geography. Combining our results with W¨astlund's tractability results gives further insight in the complexity landscape of trick-taking card games. |
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19.11.2014 |
A practical public key cryptosystem probably secure against adaptive chosen ciphertext attack |
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13.11.2014 Patryk Mikos |
Algorytmiczne Aspekty Kombinatoryki A hats game puzzle and generalized covers |
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12.11.2014 Kamil Sałaś |
Kryptologia Simple Unpredictable Pseudo-Random Number Generator |
| References:L. Blum, M. Blum, M. Shub, A Simple Unpredictable Pseudo-Random Number Generator, SIAM Journal on Computing 15(2) pp. 364--383 |
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12.11.2014 29.10.2014,Adam Polak |
Informatyka Teoretyczna On treewidth parametrization of nonpreemptive multicoloring problem |
| In the multicoloring problem we are given a graph in which every vertex has some nonnegative integer demand. We have to assign to each vertex a set of colors of the size of the demand of this vertex, in such a way that the sets of any two neighboring vertices are disjoint. In the nonpreemptive version of the problem each set of colors has to be an interval of the natural numbers. The goal is either to minimize the sum of the assigned colors, or to minimize the number of different colors used. In this talk we will discuss the fixed parameter tractability of both these problems when parametrized by the treewidth of the input graph and the maximum demand, the treewidth and the number of different demands, and the treewidth itself. |
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06.11.2014 Dorota Kapturkiewicz |
Algorytmiczne Aspekty Kombinatoryki Monotone paths in bounded degree graphs |
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05.11.2014 Bartłomiej Ryniec |
Podstawy Informatyki Social Networks with Competing Products by Krzysztof Apt and Evangelos Markakis |
| We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives. We characterize social networks for which adoption of a product by the whole network is possible (respectively necessary) and the ones for which a unique outcome is guaranteed. These characterizations directly yield polynomial time algorithms that allow us to determine whether a given social network satisfies one of the above properties. We also study algorithmic questions for networks without unique outcomes. We show that the problem of determining whether a final network exists in which all nodes adopted some product is NP-complete. In turn, we also resolve the complexity of the problems of determining whether a given node adopts some (respectively, a given) product in some (respectively, all) network(s). Further, we show that the problem of computing the minimum possible spread of a product is NP-hard to approximate with an approximation ratio better than W(n), in contrast to the maximum spread, which is efficiently computable. Finally, we clarify that some of the above problems can be solved in polynomial time when there are only two products. |
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04.11.2014 Jakub Cieśla |
Optymalizacja Kombinatoryczna Finding All Maximally-Matchable Edges in a Bipartite Graph |
| We consider the problem of finding all maximally-matchable edges in a bipartite graph G = (V, E), i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this computation in linear time O(n + m) (where n = |V| and m = |E|). Hence, the time complexity of finding all maximally-matchable edges reduces to that of finding a single maximum matching. References:T. Tassa, Finding all maximally-matchable edges in a bipartite graph, Theoret. Comput. Sci. 423 (2012), 50–58. |
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29.10.2014 Maciej Bendkowski |
Podstawy Informatyki INFINITE TIME TURING MACHINES AND AN APPLICATION TO THE HIERARCHY OF EQUIVALENCE RELATIONS ON THE REALS by SAMUEL COSKEY AND JOEL DAVID HAMKINS |
| We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $Delta^1_2$ in a satisfying way. |
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28.10.2014 Marcin Ziemiński |
Optymalizacja Kombinatoryczna Perfect Matchings in O(n log n) Time in Regular Bipartite Graphs |
| In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n log n) time (both in expectation and with high probability). The algorithm performs an appropriately truncated random walk on a modified graph to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (non-adaptive) uniform sampling established in earlier work. We also show that randomization is crucial for obtaining o(nd) time algorithms by establishing an (nd) lower bound for any deterministic algorithm. References:A. Goel, M. Kapralov, S. Khanna, Perfect matchings in O(n log n) time in regular bipartite graphs, Proceedings of the 2010 ACM International Symposium on Theory of Computing (STOC'10), 39–46, ACM, New Yo |
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22.10.2014 Grzegorz Gutowski |
Informatyka Teoretyczna Open Problem Session |
| A few interesting and promising open problems, including, but not limited to: * Coloring triangle-free graphs, * Complexity of graph classes defined by forbidden ordered subgraphs, * Reconstructing random strings from random substrings, * Scheduling multiprocessor jobs, * Storing small sets in just a few bits, * Colorful homomorphisms of planar graphs, * Domination games. |
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22.10.2014 Łukasz Lachowski |
Podstawy Informatyki Typed combinatory logic by Henk Barendregt |
| Basic properties of typed combinatory logic |
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21.10.2014 Patryk Mikos |
Optymalizacja Kombinatoryczna Maximum Matching in Regular and Almost Regular Graphs |
| An O(n^2*log(n))-time algorithm that finds a maximum matching in a regular graph with n vertices. More generally, the algorithm runs in O(r*n^2 log n) time if the difference between the maximum degree and the minimum degree is less than r. This running time is faster than applying the fastest known general matching algorithm that runs in O(√nm)-time for graphs with m edges, whenever m = ω(rn1.5 log n). References:R. Yuster, Maximum matching in regular and almost regular graphs, Algorithmica 66 (2013), no. 1, 87–92. |
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21.10.2014 Adam Gągol |
Algorytmiczne Aspekty Kombinatoryki Ternary pattern avoidance in partial words |
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15.10.2014 Michał Staromiejski |
Kryptologia On Shoup's lower bound technique for generic algorithms for discrete logarithm problem |
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15.10.2014 Łukasz Lachowski |
Podstawy Informatyki Combinatrory Logic by Henk Barendregt |
| Basic properties of combinatory logic |
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14.10.2014 07.10.2014,Bartłomiej Bosek |
Optymalizacja Kombinatoryczna Incremental algorithm on bipartite graphs |
| The talk presents the jont work of Bartłomiej Bosek, Darek Leniowski, Piotr Sankowski, and Anna Zych. We investigated the problem of maintaining maximum size matchings in incremental bipartite graphs. In this problem a bipartite graph G between n clients and n servers is revealed online. The clients arrive in an arbitrary order and request to be matched to a subset of servers. In our model we allow the clients to switch between servers and want to maximize the matching size between them, i.e., after a client arrives we find an augmenting path from a client to a free server. Our goals in this model are twofold. First, we want to minimize the number of times clients are reallocated between the servers. Second, we want to give fast algorithms that recompute such reallocation. References:Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, Anna Zych. Online bipartite matching in offline time. In Proceedings of the 55th Symposium on Foundations of Computer Science, FOCS14, pp. 384-393, 2014. |
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08.10.2014 Jakub Brzeski |
Kryptologia Continued Fractions: theory and applications |
| In the talk we focus on the most important (and interesting) properties of the continued fractions together with examples of their applications. |
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08.10.2014 Łukasz Lachowski |
Podstawy Informatyki Combinatrory Logic by Henk Barendregt |
| Basic properties of combinatory logic |
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12.06.2014 Andrzej Głuszyński |
Kryptologia Factoring with General Number Field Sieve |
| The number field sieve (NFS) is the most efficient classical algorithm known for factoring integers larger than 100 digits. Heuristically, its complexity for factoring an integer n is of the form L[1/3, (64/9)^{1/3}]. The principle of the NFS can be understood as an improvement to the simpler rational and quadratic sieve which base on searching for smooth numbers. NFS had some spectacular successes with integers in certain special forms, most notably the factorization of the 155 decimal digit ninth Fermat number F9 = 2^512 + 1. References:Peter Stevenhagen, The number field sieve. Algorithmic Number Theory, MSRI Publication Vol. 44, 2008 Carl Pomerance, The number field sieve, Proceedings of Symposis in Applied Mathematics, Vol. 48. 1994 |
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11.06.2014 Radosław Smyrek |
Informatyka Teoretyczna Shortest Path Problems on a Polyhedral Surface (by Atlas F. Cook IV, CarolaWenk) |
| We describe algorithms to compute edge sequences, a shortest path map, and the Fréchet distance for a convex polyhedral surface. Distances on the surface are measured by the length of a Euclidean shortest path. We describe how the star unfolding changes as a source point slides continuously along an edge of the convex polyhedral surface. We describe alternative algorithms to the edge sequence algorithm of Agarwal et al. (SIAM J. Comput. 26(6):1689–1713, 1997) for a convex polyhedral surface. Our approach uses persistent trees, star unfoldings, and kinetic Voronoi diagrams. We also show that the core of the star unfolding can overlap itself when the polyhedral surface is non-convex. References:Atlas F. Cook IV, CarolaWenk, Shortest Path Problems on a Polyhedral Surface, Algorithmica (2014) 69:58–77 |
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11.06.2014 Gabriel Fortin |
Podstawy Informatyki "The safe lambda calculus" by William Blum and C.-H. Luke Ong. |
| Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simply-typed lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of β-reduction that preserves safety. In the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the complexity of deciding beta-eta equality of two safe simply-typed terms and show that this problem is PSPACE-hard. Finally we give a game-semantic analysis of safety: We show that safe terms are denoted by P-incrementally justified strategies. Consequently pointers in the game semantics of safe λ-terms are only necessary from order 4 onwards. |
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05.06.2014 Krzysztof Kleiner |
Kryptologia Zero-knowledge proofs |
| A zero-knowledge proof is a protocol providing that one site can prove to the other that a certain statement is true without revealing any other information. We demand that if the prover knows the proof of the statement, it will be accepted, that otherwise it will get rejected with liberally high probability and that the distribution of the protocol transcript is the same (perfect zero-knowledge proofs) or computationally indistinguishable (computational zero-knowledge proofs) from the output of some probabilistic Turing Machine, which doesn't have access to any of the prover's private information. References:O. Goldreich, S. Micali, A. Wigderson, Proofs that Yield Nothing But their Validity and a Methodology of Cryptographic Protocol Design, Journal of the Association for Computing Machinery: Vol 38, No 1, July 1991, pp 691-72 |
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04.06.2014 Gabriel Fortin |
Informatyka Teoretyczna On Cutwidth Parameterized by Vertex Cover (by Marek Cygan et al.) |
| We study the CUTWIDTH problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for CUTWIDTH with running time O(2^k n^O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2^{n/2}n^O(1)) time algorithm for CUTWIDTH on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for CUTWIDTH on a graph class where the problem remains NP-complete. Additionally, we show that CUTWIDTH parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP ⊆ coNP/poly. Our kernelization lower bound contrasts with the recent results of Bodlaender et al. (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both TREEWIDTH and PATHWIDTH parameterized by vertex cover do admit polynomial kernels. References: Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh, On Cutwidth Parameterized by Vertex Cover, Algorithmica (2014) 68:940–953 |
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04.06.2014 Maciej Bendkowski |
Podstawy Informatyki On the shortest combinatory logic term without weak normalisation. |
| Combinatory logic is a formal notation for function abstraction, eliminating the notion of bound variables. In our presentation, we give proofs of non-normalization for two different S-terms, i.e. combinatory logic terms consisting of only one combinator S and term application, and present a computer-assisted proof of the least combinatory logic term without normal form. We will then discuss the decidability of normalization in the set of S-terms. |
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03.06.2014 M. Solon, P. Wójcik |
Algorytmy Randomizowane i Aproksymacyjne Polynomial coloring of 3-colorable graphs. |
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29.05.2014 Andrzej Dorobisz |
Kryptologia Breaking RSA may not be equivalent to factoring |
| This talk is based on the paper by D. Boneh and R. Venkatesan. Abstract of the paper: We provide evidence that breaking low-exponent RSA cannot be equivalent to factoring integers. We show that an algebraic reduction from factoring to breaking low-exponent RSA can be converted into an efficient factoring algorithm. Thus, in effect an oracle for breaking RSA does not help in factoring integers. Our result suggests an explanation for the lack of progress in proving that breaking RSA is equivalent to factoring. We emphasize that our results do not expose any weakness in the RSA system. |
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28.05.2014 Krzysztof Pasek |
Informatyka Teoretyczna Online Square Packing with Gravity (by S.P.Fekete, T.Kamphans, N.Schweer) |
| We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0, 1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares "hang in the air") based on ideas of shelf packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some binpacking arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a 34/13≈2.6154-competitive algorithm. References:Sándor P. Fekete, Tom Kamphans, Nils Schweer, Online Square Packing with Gravity, Algorithmica (2014) 68:1019–1044 |
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28.05.2014 Radosław Smyrek |
Podstawy Informatyki A hierarchy of hereditarily finite sets by Laurence Kirby |
| This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy. |
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27.05.2014 20.05.2014 13.05.2014 Adam Gągol |
Algorytmy Randomizowane i Aproksymacyjne Constraint Satisfaction, Packet Routing, and the Lovász Local Lemma (by Harris and Srinivasan) |
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22.05.2014 Jakub Brzeski |
Kryptologia Breaking RSA may be as difficult as factoring |
| This talk is based on the paper of Daniel R. L. Brown, who shows that if factoring is hard, then straight line programs cannot efficiently solve the low public exponent RSA problem. More precisely, no efficient algorithm can take an RSA public key as input and then output a straight line program that efficiently solves the low public exponent RSA problem for the given public key - unless factoring is easy. References:Daniel R. L. Brown, Breaking RSA May Be As Difficult As Factoring, Cryptology ePrint Archive: Report 2005/380, http://eprint.iacr.org/2005/380 |
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21.05.2014 Szymon Borak |
Informatyka Teoretyczna Competitive-reachability for special classes of graphs |
| The reachability r(D) of a directed graph D is the number of ordered pairs of distinct vertices (x, y) with a directed path from x to y. Two players maximizer and minimizer play the following game on graph G. They orient the edges of G alternately until all edges of G have been oriented. The maximizer attempts to maximize the reachability, while the minimizer attempts to minimize the reachability, of the resulting digraph. If both players play optimally, then the reachability is fixed. Competitive-reachability is a value of reachability for the optimal play on graph G. We determine the competitive-reachability for outerplanar graphs and some other special classes of graphs. |
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21.05.2014 Konrad Witaszczyk |
Podstawy Informatyki On the classification of recursive languages by John Case, Efim Kinber, Arun Sharma, and Frank Stephanc. |
A one-sided classifier for a given class of languages converges to 1 on every language from the class and outputs 0 infinitely often on languages outside the class. A two-sided classifier, on the other hand, converges to 1 on languages from the class and converges to 0 on languages outside the class. The present paper investigates one-sided and two-sided classification for classes of recursive languages. Theorems are presented that help assess the classifiability of natural classes. The relationships of classification to inductive learning theory and to structural complexity theory in terms of Turing degrees are studied. Furthermore, the special case of classification from only positive data is also investigated. |
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15.05.2014 Michał Masłowski |
Kryptologia Timing attacks |
| Fast implementations of AES and RSA use algorithms with non-constant time that attackers can affect by choosing inputs or using CPU cache. This allows recovering secret keys in local or remote attacks. This talk presents these algorithms, resulting timing attacks and mitigation techniques. References:David Brumley, Dan Boneh, Remote timing attacks are practical, https://crypto.stanford.edu/~dabo/papers/ssl-timing.pdf Paul C. Kocher, Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems, http://www.cryptography.com/public/pdf/TimingAttacks.pdf Daniel J. Bernstein, Cache-timing attacks on AES, http://cr.yp.to/papers.html#cachetiming |
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15.05.2014 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Nonrepetitive coloring of the plane |
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14.05.2014 Grzegorz Gutowski, Jakub Kozik |
Informatyka Teoretyczna Lower bound for on-line graph coloring of bipartite graphs |
| In this talk we propose a strategy for Presenter in on-line graph coloring game. The strategy constructs bipartite graphs and forces any on-line coloring algorithm to use 2 log n - 10 colors, where n is the number of vertices in the constructed graph. This is best possible up to an additive constant. References:http://arxiv.org/abs/1404.7259 |
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14.05.2014 Patryk Zaryjewski |
Podstawy Informatyki ON THE AVERAGE STATE COMPLEXITY OF PARTIAL DERIVATIVE AUTOMATA: AN ANALYTIC COMBINATORICS APPROACH by SABINE BRODA, ANTONIO MACHIAVELO, NELMA MOREIRA and ROGERIO REIS |
The partial derivative automaton is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton. By estimating the number of regular expressions that have \epsilon as a partial derivative, we compute a lower bound of the average number of mergings of states in A_pos and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in Apos. The lower bound corresponds to consider the A_pd automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the A_pd automaton for the unmarked regular expression, are very close to each other. |
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08.05.2014 Kamil Sałaś |
Kryptologia Data Encryption Standard |
| Short introduction to Data Encryption Standard. Detailed analysis of encryption function. Security: brute force and differential cryptanalysis. Overview of Triple DES. |
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07.05.2014 Maciej Gawron |
Podstawy Informatyki Constructions of asymptotically shortest k-radius sequences by Jaromczyk, Zbigniew Lonc, Mirosław Truszczynski |
Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let f_k(n) denote the length of a shortest k-radius sequence over A. We provide constructions demonstrating that (1) for every fixed k and for every fixed ε > 0, f_k(n) = 1 / 2k n^2 + O(n^{1+ε}) and (2) for every k = n^α, where α is a fixed real such that 0 < α < 1, f_k(n) = 1/2k n^2 + O(n^β ), for some β < 2 − α. Since fk(n) 1/2k n^2 − n/2k , the constructions give asymptotically optimal k-radius sequences. Finally, (3) we construct optimal 2-radius sequences for a 2p-element alphabet, where p is a prime. |
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30.04.2014 Bartłomiej Ryniec |
Podstawy Informatyki Multiparty communication complexity and very hard functions by Pavol Duriš |
A boolean function f(x_1, . . . , x_n) with x_i ∈ {0, 1}^m for each i is hard if its nondeterministic multiparty communication complexity (introduced in [in: Proceedings of the 30th IEEE FOCS, 1989, p. 428–433]), C(f), is at least nm. Note that C(f) n*m for each f(x_1, . . . , x_n) with x_i ∈ {0, 1}^m for each i. A boolean function is very hard if it is hard and its complementary function is also hard. In this paper, we show that randomly chosen boolean function f(x_1, . . . , x_n) with x_i ∈ {0, 1}^m for each i is very hard with very high probability (for n 3 and m large enough). In [in: Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science, LNCS 900, 1995, p. 350–360], it has been shown that if f(x_1, . . . , x_k , . . . , x_n) = f_1 (x_1, . . . , x_k ) · f_2(x_{k+1}, . . . , x_n), where C(f_1) > 0 and C(f_2) > 0, then C(f) = C(f1) + C(f2).We prove here an analogical result: If f(x_1, . . . , x_k , . . . , x_n) = f_1(x_1, . . . , x_k ) ⊕ f_2(x_{k+1}, . . . , x_n) then DC(f) = DC(f1) + DC(f2), where DC(g) denotes the deterministic multiparty communication complexity of the function g and "⊕" denotes the parity function. |
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24.04.2014 Wojciech Lubawski |
Algorytmiczne Aspekty Kombinatoryki Rota basis conjecture for sparse paving matroids |
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23.04.2014 Maciej Gazda Eindhoven University of Technology |
Podstawy Informatyki Zielonka's Recursive Algorithm for Parity Games |
| Parity games are infinite duration, two player games played on a finite directed graph. Vertices of the graph are labelled with natural numbers (called priorities) and the winning condition is determined by the parity of the most significant (typically maximal) priority encountered inifnitely often. The games are memoryless determined, moreover, the problem of finding the winning partition of a given game belongs to both NP and coNP complexity classes. On the other hand, no polynomial algorithm solving parity games has been found (the best one due to Jurdziński, Paterson and Zwick has subexponential running time with sqrt(n) in the exponent). In my talk, I will give a brief introduction to this intriguing computational problem, and then focus on one of the earliest and simplest solving algorithms, namely Zielonka's recursive algorithm. Even though its worst-case running time is not particularly impressive as compared to more sophisticated solvers, the experimental study of Friedmann and Lange has shown that in practice it works very well. In order to understand why it is the case, in our recent work with Tim Willemse we have analysed the performance of two variants of the algorithm (standard and optimised) on certain subclasses of parity games (dull, weak, and solitaire). Moreover, we have provided a tighter lower bound on its worst-case running time. |
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17.04.2014 Szymon Policht |
Kryptologia Stream ciphers |
| Stream ciphers are one of the most important branches of private-key cryptography. They offer strong security, combined with high speed and ease of implementation. In this talk, we define them and discuss ways to convert block ciphers to stream ones. Additionally, we introduce a powerful way of creating such ciphers - linear feedback shift registers. References:Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, chapter 6 Oded Goldreich, Foundations of Cryptography vol. 2 - Basic Applications, sections 5.3.1-5.3.2 |
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16.04.2014 Arkadiusz Olek |
Informatyka Teoretyczna Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem, (by M.Knauer, J.Spoerhase) |
| We examine the problem of determining a spanning tree of a given graph such that the number of internal nodes is maximum. The best approximation algorithm known so far for this problem is due to Prieto and Sloper and has a ratio of 2. For graphs without pendant nodes, Salamon has lowered this factor to 7/4 by means of local search. However, the approximative behaviour of his algorithm on general graphs has remained open. In this paper we show that a simplified and faster version of Salamon's algorithm yields a 5/3-approximation even on general graphs. In addition to this, we investigate a node weighted variant of the problem for which Salamon achieved a ratio of 2·Δ(G)−3. Extending Salamon's approach we obtain a factor of 3+ε for any ε>0. We complement our results with worst case instances showing that our analyses are tight. References:Martin Knauer, Joachim Spoerhase, Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem, Algorithmica, DOI 10.1007/s00453-013-9827-7 |
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16.04.2014 Agnieszka Łupińska |
Podstawy Informatyki Efficient Bracket Abstraction Using Iconic Representations for Combinators by Antoni Diller |
| Some fundamental properties of a new uni-variate bracket abstraction algorithm employing a string representation for combinators are established. In particular, if the input term has length n, where n > 1, the algorithm is called fewer than n times to produce the abstract. Furthermore, the space required to store the abstract, in the worst case, is of the order O(n). This algorithm also has a number of features that make it worthy of further attention. When it is used to abstract a variables from an input term of length n, where n > 1, fewer than an new combinators are introduced into the abstract. However, the total size of the string representations of these combinators grows quadratically in the number of variables abstracted and the space required to store the abstract, in the worst case, is of the order O(a^2 n). Fortunately, a closely related single-sweep, multi-variate algorithm exists, using an array representation for combinators, which produces an abstract whose storage requirement, in the worst case, is of the order O(an). |
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15.04.2014 Grzegorz Gutowski, Jakub Kozik |
Algorytmy Randomizowane i Aproksymacyjne Lower bounds for on-line graph colorings (cont.) |
| (join work with P. Micek and X. Zhu) |
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10.04.2014 Anna Dymek |
Kryptologia Pseudorandom generators |
| Many encryption techniques use "random variables", and proofs of their correctness are based on the low probability of guessing the value of that "random variables". For that we need random generators, or so called "pseudo-random generators", which give us values indistinguishable from truly random ones. In the talk we define pseudo-random generators, discuss their existence and describe their relations with other problems. References:O. Goldreich, Foundations of Cryptography vol. 1 - Basic Techniques, chapter 3 |
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10.04.2014 Wojciech Lubawski |
Algorytmiczne Aspekty Kombinatoryki Graph arboricity and matroids on-line |
| Kolorowanie krawędzi grafu nazwiemy poprawnym, jeśli zbiory jednokolorowe nie zawierają cykli. Wzorując się na przykładach z kolorowania wierzchołków grafu, rozważymy kilka różnych dwuosobowych gier, w których prezenter ujawnia część informacji o grafie, jak na przykład: należenie danej krawędzi do grafu, listę kolorów dozwolonych dla danej krawędzi grafu, zbiór krawędzi na których można użyć danego koloru itp., a algorytm ma za zadanie doprowadzić do poprawnego pokolorowania wszystkich krawędzi grafu. Liczbę kolorów zapewniającą algorytmowi strategię wygrywającą powiążemy z najmniejszą liczbą kolorów potrzebną do poprawnego pokolorowania off-line krawędzi grafu. Otrzymane wyniki uogólnimy do matroidów. |
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09.04.2014 Adam Polak |
Informatyka Teoretyczna A Generalization of the Convex Kakeya Problem, (by Ahn et al.) |
| Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G. References:Hee-Kap Ahn, SangWon Bae, Otfried Cheong, Joachim Gudmundsson, Takeshi Tokuyama, Antoine Vigneron, A Generalization of the Convex Kakeya Problem, Algorithmica, DOI 10.1007/s00453-013-9831-y |
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09.04.2014 Aleksandra Piktus |
Podstawy Informatyki Improved constructions of quantum automata by Andris Ambainis, Nikolajs Nahimovs |
We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use 4/epsion log (2p) states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of log p than the previously known construction. Our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction. |
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08.04.2014 01.04.2014, |
Algorytmy Randomizowane i Aproksymacyjne Cancelled (all participants are invited to the lecture of prof. A. Ruciński.) |
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03.04.2014 Michał Farnik |
Algorytmiczne Aspekty Kombinatoryki Beyond the Shannon's Bound |
| Let G=(V,E) be a multigraph of maximum degree D. The edges of G can be colored with at most 3D/2 colors by Shannon's theorem. We study lower bounds on the size of subgraphs of G that can be colored with D colors. Shannon's Theorem gives a bound of D|E|/floor(3D/2). However, for D=3, Kamiński and Kowalik showed that there is a 3-edge-colorable subgraph of size at least 7|E|/9, unless G has a connected component isomorphic to K_3+e (a K_3 with an arbitrary edge doubled). We extend this line of research by showing that G has a D-edge colorable subgraph with at least D|E|/(floor(3D/2)-1) edges, unless D is even and G contains D/2 K_3 or D is odd and G contains (D-1)/2 K_3+e. Moreover, the subgraph and its coloring can be found in polynomial time. Our results have applications in approximation algorithms for the Maximum k-Edge-Colorable Subgraph problem. For every even k>=4 we obtain a (2k+2)/(3k+2)-approximation and for every odd k>=5 we get a (2k+1)/3k-approximation. When 4<= k<= 13 this improves over earlier algorithms due to Feige et al. This is joint work with Łukasz Kowalik and Arkadiusz Socała. |
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02.04.2014 Łukasz Janiszewski |
Podstawy Informatyki Exploiting independent subformulas: A faster approximation scheme for #k-SAT by Manuel Schmitt , Rolf Wanka |
| We present an improvement on Thurley's recent randomized approximation scheme for #k-SAT where the task is to count the number of satisfying truth assignments of a Boolean function Φ given as an n-variable k-CNF. We introduce a novel way to identify independent substructures of Φ and can therefore reduce the size of the search space considerably. Our randomized algorithm works for any k. For #3-SAT, it runs in time O(ε−2 · 1.51426n), for #4-SAT, it runs in time O(ε−2 · 1.60816n), with error bound ε. |
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27.03.2014 Michał Staromiejski |
Kryptologia Bezout theorem and associativity of addition on elliptic curves |
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27.03.2014 Michał Masłowski |
Algorytmiczne Aspekty Kombinatoryki Maximizing the number of nonnegative subsets |
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26.03.2014 Jakub Adamek |
Informatyka Teoretyczna A Universal Randomized Packet Scheduling Algorithm (by Łukasz Jeż) |
| We give a memoryless scale-invariant randomized algorithm REMIX for Packet Scheduling that is e/(e−1)-competitive against an adaptive adversary. REMIX unifies most of previously known randomized algorithms, and its general analysis yields improved performance guarantees for several restricted variants, including the s-bounded instances. In particular, REMIX attains the optimum competitive ratio of 4/3 on 2-bounded instances. Our results are applicable to a more general problem, called Item Collection, in which only the relative order between packets' deadlines is known. REMIX is the optimal memoryless randomized algorithm against adaptive adversary for that problem References:Łukasz Jeż, A Universal Randomized Packet Scheduling Algorithm, Algorithmica (2013) 67:498–515, DOI 10.1007/s00453-012-9700-0 |
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26.03.2014 Michał Marczyk |
Podstawy Informatyki CAP theorem |
| We will examine Gilbert and Lynch's proof of Brewer's conjecture. The latter states that it is impossible for a distributed service to be simultaneously consistent, available and partition-tolerant (for certain natural definitions of these terms). We will then consider the real-world impact of the theorem. Based on Gilbert, Lynch, "Brewer's Conjecture and the Feasibility of Consistent, Available, Partition-Tolerant Web Services". |
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19.03.2014 Seminar cancelled |
Podstawy Informatyki DAY OF FACULTY OF MATHEMATICS AND COMPUTER SCIENCE |
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18.03.2014 Grzegorz Gutowski, Jakub Kozik |
Algorytmy Randomizowane i Aproksymacyjne Lower bounds for on-line graph colorings |
| (join work with P. Micek and X. Zhu) |
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13.03.2014 Grzegorz Guśpiel |
Kryptologia Optimal Asymmetric Encryption Padding (OAEP) |
| We discuss the encryption scheme OAEP, long considered to be the first one to achieve both good performance and provable security. The latter was not obtained, however, due to a mistake in the proof. We present the scheme and the flawed proof. abstract of the paper: Given an arbitrary k-bit to k-bit trapdoor permutation f and a hash function, we exhibit an encryption scheme for which (i) any string x of length slightly less than k bits can be encrypted as f(r_x), where r_x is a simple probabilistic encoding of x depending on the hash function; and (ii) the scheme can be proven semantically secure assuming the hash function is "ideal." Moreover, a slightly enhanced scheme is shown to have the property that the adversary can create ciphertexts only of strings for which she "knows" the corresponding plaintexts - such a scheme is not only semantically secure but also non-malleable and secure against chosen-ciphertext attack. References:M. Bellare, P. Rogaway, Optimal Asymmetric Encryption - How to Encrypt with RSA, http://cseweb.ucsd.edu/~mihir/papers/oae.pdf |
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13.03.2014 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki Coloring 3-colorable graphs on-line |
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12.03.2014 Michał Dyrek |
Informatyka Teoretyczna Balanced Partitions of Trees and Applications (by A.E.Feldmann, L.Foschini) |
| We study the problem of finding the minimum number of edges that, when cut, form a partition of the vertices into k sets of equal size. This is called the k-BALANCED PARTITIONING problem. The problem is known to be inapproximable within any finite factor on general graphs, while little is known about restricted graph classes. We show that the k-BALANCED PARTITIONING problem remains APX-hard even when restricted to unweighted tree instances with constant maximum degree. If instead the diameter of the tree is constant we prove that the problem is NP-hard to approximate within n^c, for any constant c<1. If vertex sets are allowed to deviate from being equal-sized by a factor of at most 1+ε, we show that solutions can be computed on weighted trees with cut cost no worse than the minimum attainable when requiring equal-sized sets. This result is then extended to general graphs via decompositions into trees and improves the previously best approximation ratio from O(log^{3/2}(n)/ε^2) [Andreev and Räcke in Theory Comput. Syst. 39(6):929–939, 2006] to O(log n). This also settles the open problem of whether an algorithm exists for which the number of edges cut is independent of ε. References:Andreas Emil Feldmann, Luca Foschini, Balanced Partitions of Trees and Applications, Algorithmica DOI 10.1007/s00453-013-9802-3 |
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06.03.2014 Wojciech Łopata |
Kryptologia Introduction to provable security |
| We discuss several definitions of cryptosystem security as a resistance against "chosen-ciphertext" attacks, and reveal weaknesses of RSA and ElGamal encryption schemes. Then I describe Cramer-Shoup encryption, and prove that if the Decision Diffiee-Hellman Problem is hard, then Cramer-Shoup encryption is indistinguishability-secure from chosen-ciphertext attack. |
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06.03.2014 Adam Gągol |
Algorytmiczne Aspekty Kombinatoryki Application of probabilistic method in some colourings of bounded path-width graphs |
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05.03.2014 26.02.2014,Adam Gągol |
Informatyka Teoretyczna Natural proofs (by A. Razborov, S. Rudich) |
| The notion of natural proof is introduced. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in nonmonotone models fall within our definition of natural. We show, based on a hardness assumption, that natural proofs can not prove superpolynomial lower bounds for general circuits. Without the hardness assumption, we are able to show that they can not prove exponential lower bounds (for general circuits) for the discrete logarithm problem. We show that the weaker class of AC^0-natural proofs which is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser, Yao, and Hastad is inherently incapable of proving the bounds of Razborov and Smolensky. We give some formal evidence that natural proofs are indeed natural by showing that every formal complexity measure, which can prove superpolynomial lower bounds for a single function, can do so for almost all functions, which is one of the two requirements of a natural proof in our sense. References:Alexander A. Razborov, Steven Rudich, Natural proofs, Journal of Computer and System Sciences, 55(1997), 24-35 |
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05.03.2014 Robert Obryk |
Podstawy Informatyki Cryptographic Accumulators |
| A cryptographic accumulator is a less well-known cousin of a cryptographic hash function: it allows a digest of a multiset to be constructed one element at a time. One can also extend this notion in a few ways: allow removing elements already added to the digest, or provide witnesses that prove validity of operations on the digest without giving away what operations they were. This talk will present the basic notion of accumulators and their properties, give example implementations (secure under the typical assumptions) and hint at their possible uses. |
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04.03.2014 Igor Adamski |
Algorytmy Randomizowane i Aproksymacyjne Outer-string graphs are chi-bounded |
| References:Alexandre Rok, Bartosz Walczak, Outer-string graphs are chi-bounded, preprint |
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26.02.2014 Adam Polak |
Podstawy Informatyki Open problems for pattern languages |
| A pattern is a string built of terminals and variables. A language generated by a given pattern consists of words produced by substituting variables with arbitrary strings of terminals. Of course every occurrence of the same variable has to be substituted with the same string. Pattern languages were first studied in the context of machine learning but soon attracted formal languages researchers. Despite their very simple definition they have numerous interesting properties. During the seminar we will discuss several intriguing computational and structural problems involving pattern languages and their relation to the Chomsky hierarchy. Some of them were recently solved and some remain open. Papers about pattern languages: http://www.tks.informatik.uni-frankfurt.de/data/doc/dissertation.pdf http://link.springer.com/chapter/10.1007%2F3-540-56939-1_81 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.3057&rep=rep1&type=pdf http://www.sciencedirect.com/science/article/pii/S0890540109001023 http://www.tks.informatik.uni-frankfurt.de/data/doc/dlt2010.pdf http://www.comp.nus.edu.sg/~sanjay/paps/regpat.pdf http://www.sciencedirect.com/science/article/pii/S030439751300577X |
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23.01.2014 Karol Kosiński |
Algorytmiczne Aspekty Kombinatoryki A generalization of Thue freeness for partial words |
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22.01.2014 Maciej Solon |
Informatyka Teoretyczna Scheduling with an Orthogonal Resource Constraint |
| We address a scheduling problem that arises in highly parallelized environments like modern multi-core CPU/GPU computer architectures where simultaneously active jobs share a common limited resource, e.g., memory cache. The scheduler must ensure that the demand for the common resource never exceeds the available capacity. This introduces an orthogonal constraint to the classical minimum makespan scheduling problem. Such a constraint also arises in other contexts where a common resource is shared across machines. We study the non-preemptive case of this problem and present a (2+\epsi)-approximation algorithm which relies on the interplay of several classical and modern techniques in scheduling like grouping, job-classification, and the use of configuration-LPs. This improves upon previous bound of 3 that can be obtained by list scheduling approaches, and gets close to the (3/2−\epsi)-inapproximability bound. If the number of machines or the number of different resource requirements are bounded by a constant we obtain a polynomial time approximation scheme. References:Martin Niemeier, Andreas Wiese, Scheduling with an Orthogonal Resource Constraint, Algorithmica, DOI 10.1007/s00453-013-9829-5 |
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22.01.2014 Maciej Bendkowski |
Podstawy Informatyki An upper bound for reduction sequences in the typed lambda-calculus by H. Schwichtenberg |
| It is well known that the full reduction tree for any term of the typed λ–calculus is finite. However, it is not obvious how a reasonable estimate for its height might be obtained. Here we note that the head reduction tree has the property tha the number of its nodes with conversions bounds the length of any reduction sequence. The height of that tree, and hence also the number of its nodes, can be estimated using a technique due to Howard [3], which in turn is based on work of Sanchis [4] and Diller [1]. This gives the desired upper bound. The method of Gandy [2] can also be used to obtain a bound for the length of arbitrary reduction sequences; this is carried out in [5]. However, the bound derived here, apart from being more intelligible, is also better. |
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21.01.2014 Maciej Solon |
Algorytmy Randomizowane i Aproksymacyjne On the number of edges in families of pseudo-discs |
| References:Piotr Micek, Rom Pinchasi, On the number of edges in families of pseudo-discs, preprint |
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16.01.2014 Jaroslaw Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Three variations on the theme of Szemeredi |
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15.01.2014 Michał Sapalski |
Informatyka Teoretyczna Linear-Time Algorithms for Tree Root Problems |
| Let T be a tree on a set V of nodes. The p-th power T^p of T is the graph on V such that any two nodes u and w of V are adjacent in T^p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T , if any, such that G = T^p. Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T , if any, such that G = T^p. Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n^3) (respectively, O(n^4)) time. In this paper, we give O(n+m)-time algorithms for both problems. References:Maw-Shang Chang, Ming-Tat Ko, Hsueh-I Lu, Linear-Time Algorithms for Tree Root Problems, Algorithmica, DOI 10.1007/s00453-013-9815-y |
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15.01.2014 Konrad Witaszczyk |
Podstawy Informatyki Analytic aspects of the shuffle product by Marni Mishna, Mike Zabrocki |
| There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models D-finite generating functions. |
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14.01.2014 Maciej Bendkowski |
Algorytmy Randomizowane i Aproksymacyjne List Colouring When The Chromatic Number Is Close To the Order Of The Graph |
| References:B. Reed, B. Sudakov, List Colouring When The Chromatic Number Is Close To the Order Of The Graph, Combinatorica December 2004, Volume 25, Issue 1, pp 117-123 |
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09.01.2014 Marcin Dziaduś |
Algorytmiczne Aspekty Kombinatoryki Coloring intersection graphs of pseudo-discs |
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08.01.2014 Andrzej Dorobisz |
Informatyka Teoretyczna Linear Recognition and Embedding of Fibonacci Cubes |
| Fibonacci strings are binary strings that contain no two consecutive 1s. The Fibonacci cube Γ_h is the subgraph of the h-cube induced by the Fibonacci strings. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. We derive a new characterization of Fibonacci cubes. The characterization is the basis for an algorithm which recognizes these graphs in linear time. Moreover, a graph which was recognized as a Fibonacci cube can be embedded into a hypercube using Fibonacci strings within the same time bound. References:Aleksander Vesel, Linear Recognition and Embedding of Fibonacci Cubes, Algorithmica, DOI 10.1007/s00453-013-9839-3 |
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08.01.2014 Michał Bejda |
Podstawy Informatyki Generalized satisfability problems: minimal elements and phase transitions by Nadia Creignoua, Herve Daud |
| We develop a probabilistic model on the generalized satisfability problems defned by Schaefer (in: Proceedings of the 10th STOC, San Diego, CA, USA, Association for Computing Machinery, New York, 1978, pp. 216–226) for which the arity of the constraints is fixed in order to study the associated phase transition. We establish new results on minimal elements associated with such generalized satis"ability problems. These results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfability. We first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio. Our framework allows one to get these bounds in a uniform way, in particular, we obtain a lower bound proportional to the number of variables for k-SAT without analyzing any algorithm. Finally, we reveal the seed of coarseness for the phase transition of generalized satisfability: 2-XOR-SAT. |
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07.01.2014 Aneta Pawłowska |
Algorytmy Randomizowane i Aproksymacyjne Three Topics in Online List Coloring |
| References:J. Carraher, S. Loeb, T. Mahoney, G. Puleo, M. Tsai, D.B. West,Three Topics in Online List Coloring, preprint |
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19.12.2013 Jan Volec (University of Warwick) |
Algorytmiczne Aspekty Kombinatoryki Compactness and finitely forcible graphons |
| Graphons are limit objects that are associated with convergent sequences of graphs. Problems from extremal combinatorics led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible graphons. In 2011, Lovasz and Szegedy asked several questions about the complexity of the topological space of so-called typical vertices of a finitely forcible graphon can be. In particular, they conjectured that the space is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space of typical vertices is not compact. In fact, our construction actually provides an example of a finitely forcible graphon with the space which is even not locally compact. This is a joint work with Roman Glebov and Dan Kral. |
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18.12.2013 Bartłomiej Ryniec |
Informatyka Teoretyczna Preprocess, Set, Query! |
| Thorup and Zwick (J.ACM 52(1):1–24, 2005 and STOC'01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k ≥ 1 it is possible to preprocess the graph in O˜(m n^{1/k}) time and generate a compact data structure of size O(k n^{1+1/k}). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k−1 in O(k) time. For k=2 this gives an oracle of O(n^{1.5}) size that produces in constant time estimated distances with stretch 3. Recently, Patrascu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n^{5/3}) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs.We present a data structure that produces estimated distances with 1+ε stretch. The size of the data structure is O(n m^{1−ε'}) and the query time is O˜(m^{1−ε'}). Using it for sparse unweighted graphs we can get a data structure of size O(n^{1.87}) that can supply in O(n^{0.87}) time estimated distances with multiplicative stretch 1.75. References:Ely Porat, Liam Roditty, Preprocess, Set, Query! Algorithmica (2013) 67:516–528 |
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18.12.2013 Łukasz Janiszewski |
Podstawy Informatyki . Exploiting independent subformulas: A faster approximation scheme for #k-SAT by Manuel Schmitt , Rolf Wanka |
| We present an improvement on Thurley's recent randomized approximation scheme for #k-SAT where the task is to count the number of satisfying truth assignments of a Boolean function Φ given as an n-variable k-CNF. We introduce a novel way to identify independent substructures of Φ and can therefore reduce the size of the search space considerably. Our randomized algorithm works for any k. For #3-SAT, it runs in time O(ε−2 · 1.51426n), for #4-SAT, it runs in time O(ε−2 · 1.60816n), with error bound ε. |
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17.12.2013 10.12.2013,Damian Królik |
Algorytmy Randomizowane i Aproksymacyjne Sum-paintability of generalized theta-graphs |
| References:J. Carraher, T. Mahoney, G.J. Puleo, D.B. West , Sum-paintability of generalized theta-graphs, preprint |
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11.12.2013 Agnieszka Dymel |
Informatyka Teoretyczna Online Coloring of Bipartite Graphs with and without Advice |
| In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least \floor{1.13746·log_2(n) − 0.49887} colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of \floor{log_2(n)}+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that O(√n) advice bits are sufficient for coloring any bipartite graph on n vertices with at most \ceil{log_2(n)} colors. References:Maria Paola Bianchi, Hans-Joachim Böckenhauer, Juraj Hromkovic, Lucia Keller, Online Coloring of Bipartite Graphs with and without Advice Algorithmica, DOI 10.1007/s00453-013-9819-7 |
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11.12.2013 Michał Dyrek |
Podstawy Informatyki Boundary properties of the satisfiability problems by Vadim Lozin , Christopher Purcell |
| The satisfiability problem is known to be NP-complete in general and for many restricted instances, such as CNF formulas with at most 3 variables per clause and at most 3 occurrences per variable, or planar formulas. The latter example refers to graphs representing satisfiability instances. These are bipartite graphs with vertices representing clauses and variables, and edges connecting variables to the clauses containing them. Finding the strongest possible restrictions under which the problem remains NP-complete is important for at least two reasons. First, this can make it easier to establish the NP completeness of new problems by allowing easier transformations. Second, this can help clarify the boundary between tractable and intractable instances of the problem. In this paper, we address the second issue and reveal the first boundary property of graphs representing satisfiability instances. |
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04.12.2013 Sebastian Syta |
Informatyka Teoretyczna Online Unweighted Knapsack Problem with Removal Cost |
| We study the online unweighted knapsack problem with removal cost. The input is a sequence of items u_1,u_2,...,u_n, each of which has a size and a value, where the value of each item is assumed to be equal to the size. Given the ith item u_i, we either put u_i into the knapsack or reject it with no cost. When u_i is put into the knapsack, some items in the knapsack are removed with removal cost if the sum of the size of u_i and the total size in the current knapsack exceeds the capacity of the knapsack. Here the removal cost means a cancellation charge or disposal fee. Our goal is to maximize the profit, i.e., the sum of the values of items in the last knapsack minus the total removal cost occurred. We consider two kinds of removal cost: unit and proportional cost. For both models, we provide their competitive ratios. Namely, we construct optimal online algorithms and prove that they are best possible. References:Xin Han, Yasushi Kawase, Kazuhisa Makino, Online Unweighted Knapsack Problem with Removal Cost, Algorithmica DOI 10.1007/s00453-013-9822-z |
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04.12.2013 Przemysław Jedynak |
Podstawy Informatyki A Myhill-Nerode theorem for automata with advice by Alex Kruckman, Sasha Rubin, John Sheridan |
| An automaton with advice is a finite state automaton which has access to an additional fixed infinite string called an advice tape. We refine the Myhill-Nerode theorem to characterize the languages of finite strings that are accepted by automata with advice. We do the same for tree automata with advice. |
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28.11.2013 21.11.2013,Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki The weak 3-flow conjecture and the weak circular flow conjecture |
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27.11.2013 Michał Bejda |
Informatyka Teoretyczna Data Structures on Event Graphs |
| We investigate the behavior of data structures when the input and operations are generated by an event graph. This model is inspired by Markov chains. We are given a fixed graph G, whose nodes are annotated with operations of the type insert, delete, and query. The algorithm responds to the requests as it encounters them during a (random or adversarial) walk in G. We study the limit behavior of such a walk and give an efficient algorithm for recognizing which structures can be generated. We also give a near-optimal algorithm for successor searching if the event graph is a cycle and the walk is adversarial. For a random walk, the algorithm becomes optimal. References:Bernard Chazelle, Wolfgang Mulzer, Data Structures on Event Graphs, Algorithmica DOI 10.1007/s00453-013-9838-4 |
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27.11.2013 Adam Polak |
Podstawy Informatyki On the satisfiability threshold and clustering of solutions of random 3-SAT formulas, by Elitza Maneva, Alistair Sinclair |
We study the structure of satisfying assignments of a random 3-Sat formula. In particular, we show that a random formula of density 4:453 almost surely has no non-trivial ``core'' assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random Sat. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-Sat with k 9 [D. Achlioptas, F. Ricci-Tersenghi, On the solution-space geometry of random constraint satisfaction problems, in: Proc. 38th ACM Symp. Theory of Computing, STOC, 2006, pp. 130 139]. Our result implies that either this does not hold for 3-Sat, or the threshold density for satisfiability in 3-Sat lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method |
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26.11.2013 Wojciech Łopata |
Algorytmy Randomizowane i Aproksymacyjne Conflict-Free Colourings of Uniform Hypergraphs With Few Edges |
| References:A.V. Kostochka, M. Kumbhat, T. Łuczak, Conflict-Free Colourings of Uniform Hypergraphs With Few Edges, Combinatorics, Probability and Computing Volume 21 Issue 4, July 2012 |
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20.11.2013 Damian Krolik |
Informatyka Teoretyczna Parameterized Analysis of Paging and List Update Algorithms |
| It is well-established that input sequences for paging and list update have locality of reference. In this paper we analyze the performance of algorithms for these problems in terms of the amount of locality in the input sequence. We define a measure for locality that is based on Denning's working set model and express the performance of well known algorithms in terms of this parameter. This explicitly introduces parameterized-style analysis to online algorithms. The idea is that rather than normalizing the performance of an online algorithm by an (optimal) offline algorithm, we explicitly express the behavior of the algorithm in terms of two more natural parameters: the size of the cache and Denning's working set measure. This technique creates a performance hierarchy of paging algorithms which better reflects their experimentally observed relative strengths. It also reflects the intuition that a larger cache leads to a better performance. We also apply the parameterized analysis framework to list update and show that certain randomized algorithms which are superior to MTF in the classical model are not so in the parameterized case, which matches experimental results. References:Reza Dorrigiv, Martin R. Ehmsen, Alejandro López-Ortiz, Parameterized Analysis of Paging and List Update Algorithms, Algorithmica, DOI 10.1007/s00453-013-9800-5 |
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20.11.2013 Andrzej Dorobisz |
Podstawy Informatyki Regular Languages Accepted by Quantum Automata by Alberto Bertoni and Marco Carpentieri |
| In this paper we analyze some features of the behaviour of quantum automata. In particular we prove that the class of languages recognized by quantum automata with isolated cut point is the class of reversible regular languages. As a more general result, we give a bound on the inverse error that implies the regularity of the language accepted by a quantum automaton |
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14.11.2013 Robert Obryk |
Algorytmiczne Aspekty Kombinatoryki Network routing as a multiparty game with asynchronous moves |
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13.11.2013 Maciej Bendkowski |
Informatyka Teoretyczna Analyses of Cardinal Auctions |
| We study cardinal auctions for selling multiple copies of a good, in which bidders specify not only their bid or how much they are ready to pay for the good, but also a cardinality constraint on the number of copies that will be sold via the auction. We perform first known Price of Anarchy type analyses with detailed comparison of the classical Vickrey-Clarke-Groves (VCG) auction and one based on minimum pay property (MPP) which is similar to Generalized Second Price auction commonly used in sponsored search. Without cardinality constraints, MPP has the same efficiency (total value to bidders) and at least as much revenue (total income to the auctioneer) as VCG; this also holds for certain other generalizations of MPP (e.g., prefix constrained auctions, as we show here). In contrast, our main results are that, with cardinality constraints, (a) equilibrium efficiency of MPP is 1/2 of that of VCG and this factor is tight, (b) in equilibrium MPP may collect as little as 1/2 the revenue of VCG. References:Mangesh Gupte, Darja Krushevskaja, S. Muthukrishnan, Analyses of Cardinal Auctions, Algorithmica DOI 10.1007/s00453-013-9832-x |
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13.11.2013 Kamil Jarosz |
Podstawy Informatyki On the strongly generic undecidability of the Halting Problem by Alexander Rybalov |
It has been shown in [J.D. Hamkins, A. Miasnikov, The halting problem is decidable on a set of asymptotic probability one, Notre Dame J. Formal Logic 47(4) (2006) 515–524] that the classical Halting Problem for Turing machines with one-way tape is decidable on a "large" set of Turing machines (a so-called generic set). However, here we prove that the Halting Problem remains undecidable when restricted to an arbitrary "very large" set of Turing machines (a so-called strongly generic set). Our proof is independent of a Turing machine model. |
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12.11.2013 Tomasz Krawczyk, Bartosz Walczak |
Algorytmy Randomizowane i Aproksymacyjne Coloring subtree overlap graphs with O(log lgo n) colors. |
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07.11.2013 Karol Kosiński |
Algorytmiczne Aspekty Kombinatoryki On some properties of (strongly) non-repetitive sequences |
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06.11.2013 Karol Różycki |
Informatyka Teoretyczna Oblivious Algorithms for the Maximum Directed Cut Problem |
| We introduce a special family of randomized algorithms for Max DICUT that we call oblivious algorithms. Let the bias of a vertex be the ratio between the total weight of its outgoing edges and the total weight of all its edges. An oblivious algorithm selects at random in which side of the cut to place a vertex v, with probability that only depends on the bias of v, independently of other vertices. The reader may observe that the algorithm that ignores the bias and chooses each side with probability 1/2 has an approximation ratio of 1/4, whereas no oblivious algorithm can have an approximation ratio better than 1/2 (with an even directed cycle serving as a negative example). We attempt to characterize the best approximation ratio achievable by oblivious algorithms, and present results that are nearly tight. The paper also discusses natural extensions of the notion of oblivious algorithms, and extensions to the more general problem of Max 2-AND. References:Uriel Feige, Shlomo Jozeph, Oblivious Algorithms for the Maximum Directed Cut Problem, Algorithmica DOI 10.1007/s00453-013-9806-z |
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06.11.2013 Michał Masłowski |
Podstawy Informatyki The Halting Problem Is Decidable on a Set of Asymptotic Probability One by Joel David Hamkins and Alexei Miasnikov |
| The halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational models. |
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05.11.2013 Grzegorz Guśpiel |
Algorytmy Randomizowane i Aproksymacyjne On the construction of 3-chromatic hypergraphs with few edges. |
| References:Heidi Gebauer, On the construction of 3-chromatic hypergraphs with few edges, JCTA 120 (2013) |
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30.10.2013 Igor Adamski |
Informatyka Teoretyczna Linked Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space |
| The dynamic trie is a fundamental data structure with applications in many areas of computer science. This paper proposes a new technique for maintaining a dynamic trie T of size at most 2^w nodes under the unit-cost RAM model with a fixed word size w. It is based on the idea of partitioning T into a set of linked small tries, each of which can be maintained efficiently. Our method is not only space-efficient, but also allows the longest common prefix between any query pattern P and the strings currently stored in T to be computed in o(|P|) time for small alphabets, and allows any leaf to be inserted into or deleted from T in o(log|T|) time. To demonstrate the usefulness of our new data structure, we apply it to LZ-compression. Significantly, we obtain the first algorithm for generating the LZ78 encoding of a given string of length n over an alphabet of size σ in sublinear (o(n)) time and sublinear (o(n log σ) bits) working space for small alphabets (σ = 2^{o(log n \cdot \frac{log log log n}{(log log n)^2})). Moreover, the working space for our new algorithm is asymptotically less than or equal to the space for storing the output compressed text, regardless of the alphabet size. References:Jesper Jansson, Kunihiko Sadakane, Wing-Kin Sung, Linked Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space, Algorithmica DOI 10.1007/s00453-013-9836-6 |
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30.10.2013 Agnieszka Łupińska |
Podstawy Informatyki Design and Implementation of a Parallel Priority Queue on Many-core Architectures by Xi He, Dinesh Agarwal, and Sushil K. Prasad |
| An efficient parallel priority queue is at the core of the effort in parallelizing important non-numeric irregular computations such as discrete event simulation scheduling and branch-and-bound algorithms. GPGPUs can provide powerful computing platform for such non-numeric computations if an efficient parallel priority queue implementation is available. In this paper, aiming at fine-grained applications, we develop an efficient parallel heap system employing CUDA. To our knowledge, this is the first parallel priority queue implementation on manycore architectures, thus represents a breakthrough. By allowing wide heap nodes to enable thousands of simultaneous deletions of highest priority items and insertions of new items, and taking full advantage of CUDA's data parallel SIMT architecture, we demonstrate up to 30-fold absolute speedup for relatively finegrained compute loads compared to optimized sequential priority queue implementation on fast multicores. Compared to this, our optimized multicore parallelization of parallel heap yields only 2-3 fold speedup for such fine-grained loads. This parallelization of a tree-based data structure on GPGPUs provides a roadmap for future parallelizations of other such data structures. |
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24.10.2013 Wiktor Kuropatwa |
Algorytmiczne Aspekty Kombinatoryki Distortion-colouring of cubic bipartite multigraphs |
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23.10.2013 Wojciech Łopata |
Informatyka Teoretyczna An Algorithmic Characterization of Polynomial Functions over Z_{p^n} |
| We consider polynomial representability of functions defined over Z_{p^n}, where p is a prime and n is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to determine whether a given function over Z_{p^n} is polynomially representable or not, (ii) finds the polynomial if it is polynomially representable. The previous characterizations given by Kempner (Trans. Am. Math. Soc. 22(2):240–266, 1921) and Carlitz (Acta Arith. 9(1), 67–78, 1964) are existential in nature and only lead to an exhaustive search method, i.e. algorithm with complexity exponential in size of the input. Our characterization leads to an algorithm whose running time is linear in size of input. We also extend our result to the multivariate case. References:Ashwin Guha, Ambedkar Dukkipati, An Algorithmic Characterization of Polynomial Functions over Z_{p^n}, Algorithmica DOI 10.1007/s00453-013-9799-7 |
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23.10.2013 Michał Marczyk |
Podstawy Informatyki Consistency in distributed systems, part II: avoiding synchronization with CRDTs (based on a paper by Shapiro, Preguiça, Baquero, Zawirski) |
| In this closing part of a two-part series we will consider CRDTs, a systematic approach to eventual consistency. We will examine both the state-based and the operation-based approach, with some concrete examples. Finally, we will return to our motivating discussion from part I in considering how a system may incorporate both an eventually consistent data store and a limited dose of consensus to achieve excellent functional guarantees in a distributed setting. Key source: Shapiro, Preguiça, Baquero, Zawirski, "A comprehensive study of Convergent and Commutative Replicated Data Types", http://hal.inria.fr/inria-00555588/ |
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17.10.2013 Jakub Kozik |
Algorytmiczne Aspekty Kombinatoryki Random Greedy Coloring of Uniform Hypergraphs |
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16.10.2013 Jerzy MarcinkowskiUniversity of Wrocław |
Informatyka Teoretyczna Finite Controllability and Bounded Derivation Depth |
| FC (Finite controllability) and BDD (the Bounded Derivation Depth property) are two properties of Datalog/TGD programs. BDD is equivalent to Positive First Order rewritability -- the very useful property that allows us to use (all the optimizations of) DBMS in order to compute the certain answers to queries in the presence of a theory. Finite Controllability of a theory T means that if the certain answer to a query Q, for a database instance D , in the presence of T is 'no' then this 'no' is never a result of an unnatural assumption that the counterexample can be infinite. We conjecture that for any theory T the property BDD implies FC. We prove this conjecture for the case of binary signatures. References:Tomasz Gogacz, Jerzy Marcinkowski: On the BDD/FC conjecture. Proceedings of PODS 2013 (the 32nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems) |
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16.10.2013 Marek Markiewicz |
Podstawy Informatyki Cellular Automata on a Toeplitz Space. |
| Toeplitz Space is a set of regular quasi-periodic bi-infinite words over a finite alphabet with at least two different letters endowed with a Besicovitch metric. It is an invariant set for every Cellular Automaton. During the talk I will present some properties of this space and I will discuss how CA behave on it. I will also present an idea of examining the continuity of evolution of a CA and show some very basic results in this topic. |
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14.10.2013 W. Hugh WoodinUC Berkeley |
Informatyka Teoretyczna The Continuum Hypothesis and the search for Mathematical Infinitynew place and date: Oct 14, 2013, 16:00,Polska Akademia Umiejętności, Kraków, Sławkowska 17 |
| By middle of the 20th century the problem of the Continuum Hypothesis was widely regarded as one of the prominent problems in all of Mathematics. Remarkably, this question cannot be solved on the basis of the basic principles (these are the ZFC axioms) on which the entire subject is based. This discovery of Cohen, 50 years ago this summer, is arguably one of the major discoveries in the history of modern Mathematics. But does this mean that the question of the Continuum Hypothesis has no answer? Any "solution" must involve the adoption of new principles but which principles should one adopt? Alternatively, perhaps the correct assessment of Cohen's discovery is that the entire enterprise of the mathematical study of Infinity is ultimately doomed because the entire subject is simply a human fiction without any true foundation. In this case, any "solution" to the Continuum Hypothesis is just an arbitrary (human) choice. Over the last few years a scenario has emerged by which with the addition of a single new principle not only can the problem of the Continuum Hypothesis be resolved, but so can all of the other problems which Cohen's method has been used to show are also unsolvable (and there have been many such problems). Moreover the extension of the basic (ZFC) principles by this new principle would be seen as an absolutely compelling option based on the fundamental intuitions on which the entire mathematical conception of Infinity is founded. However, this scenario critically depends upon the outcome of a single conjecture. If this conjecture is false then the entire approach, which is the culmination of nearly 50 years of research, fails or at the very least has to be significantly revised. Thus the mathematical study of Infinity has arguably reached a tipping point. But which way will it tip? |
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10.10.2013 Dawid Ireno |
Algorytmiczne Aspekty Kombinatoryki The Cinderella Game on Holes and Anti-holes |
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09.10.2013 Michał Marczyk |
Podstawy Informatyki Consistency in distributed systems, part I: achieving consensus with Raft (presenting research by Ongaro & Ousterhout) |
| In this opening part of a two-part series we will consider Raft, a new protocol for achieving consensus in a distributed system. Raft matches Paxos as far as efficiency is concerned, but is designed to be more readily understandable and more amenable to implementation without tweaks and additional optimizations. To motivate the discussion, we will briefly consider the concept of a distributed system and the circumstances in which such systems may or may not require consensus to make progress safely. Key source: Ongaro, Ousterhout, "In Search of an Understandable Consensus Algorithm" (draft, 2013-09-10), https://ramcloud.stanford.edu/wiki/download/attachments/11370504/raft.pdf |
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03.10.2013 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Fractions, Continued and Egyptian |
| I will present some problems and results on continued fractions and Egyptian fractions. |
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19.06.2013 Jean CardinalUniversité Libre de Bruxelles |
Informatyka Teoretyczna On Universal Point Sets for Planar Graphs and Related Problems |
| A set S of points in the plane is said to be n-universal if every planar graph on n vertices has a straight-line plane embedding on a subset of S. Finding the minimum size f(n) of an n-universal point set is a long-standing open problem, and the current upper and lower bounds differ by a linear factor. We will consider a lower bound technique that allowed us to prove that there is no n-universal point set of size n for any n>14. We will also describe recent results on families of planar graphs on n vertices that cannot be embedded on a common n-point set. This is a joint work with Michael Hoffmann and Vincent Kusters. |
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13.06.2013 Jakub Brzeski |
Algorytmiczne Aspekty Kombinatoryki The universal and canonically colored sequences |
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12.06.2013 Marcin Ziemiński |
Informatyka Teoretyczna DAGGER: A Scalable Index for Reachability Queries in Large Dynamic Graphs |
| With the ubiquity of large-scale graph data in a variety of application domains, querying them effectively is a challenge. In particular, reachability queries are becoming increasingly important, especially for containment, subsumption, and connectivity checks. Whereas many methods have been proposed for static graph reachability, many real-world graphs are constantly evolving, which calls for dynamic indexing. In this paper, we present a fully dynamic reachability index over dynamic graphs. Our method, called DAGGER, is a light-weight index based on interval labeling, that scales to million node graphs and beyond. Our extensive experimental evaluation on real-world and synthetic graphs confirms its effectiveness over baseline methods. References:Hilmi Yildirim, Vineet Chaoji, Mohammed J.Zaki, DAGGER: A Scalable Index for Reachability Queries in Large Dynamic Graphs, arXiv:1301.0977 |
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06.06.2013 Marcin Dziaduś |
Algorytmiczne Aspekty Kombinatoryki The Caccetta-Haggkvist Conjecture and Additive Number Theory |
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05.06.2013 Patryk Zaryjewski |
Informatyka Teoretyczna In-situ associative permuting |
| The technique of in-situ associative permuting is introduced which is an association of in-situ permuting and in-situ inverting. It is suitable for associatively permutable permutations of {1,2,...,n} where the elements that will be inverted are negative and stored in order relative to each other according to their absolute values. Let K[1...n] be an array of n integer keys each in the range [1,n], and it is allowed to modify the keys in the range [-n,n]. If the integer keys are rearranged such that one of each distinct key having the value i is moved to the i'th position of K, then the resulting arrangement (will be denoted by K^P) can be transformed in-situ into associatively permutable permutation pi^P using only logn additional bits. The associatively permutable permutation pi^P not only stores the ranks of the keys of K^P but also uniquely represents K^P. Restoring the keys from pi^P is not considered. However, in-situ associative permuting pi^P in O(n) time using logn additional bits rearranges the elements of pi^P in order, as well as lets to restore the keys of K^P in O(n) further time using the inverses of the negative ranks. This means that an array of n integer keys each in the range [1,n] can be sorted using only logn bits of additional space. References:A. Emre Cetin, In-situ associative permuting, arXiv:1301.2046 |
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05.06.2013 Łukasz Janiszewski |
Podstawy Informatyki Tetris is Hard, Even to Approximate by Erik D. Demaine, Susan Hohenberger and David Liben-Nowell |
| In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^{1−epsilon}, when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2−epsilon, for any epsilon>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets. |
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23.05.2013 Wojciech Lubawski |
Algorytmiczne Aspekty Kombinatoryki Lovasz's original proof of Kneser conjecture |
| In the last thirty years algebraic topology has become an important tool in combinatorics. We will show a classical proof of Kneser conjecture in which the author related n-colorability of a graph with k-connectedness of a neighbourhood simplicial complex. If time permits we will show some other applications of algebraic topology in combinatorics. |
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22.05.2013 Przemysław Derengowski |
Informatyka Teoretyczna Proper Interval Vertex Deletion |
| The NP-complete problem PROPER INTERVAL VERTEX DELETION is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 410, pp.232–243, 2010) showed that this problem can be solved in O((14k+14)^{k+1}kn^6) time. We improve this result by presenting an O(6^kkn^6) time algorithm for PROPER INTERVAL VERTEX DELETION. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw, net, tent, C_4, C_5, C_6}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves PROPER INTERVAL VERTEX DELETION on {claw, net, tent, C_4, C_5, C_6}-free graphs in O(n+m) time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of PROPER INTERVAL VERTEX DELETION. References:Pim van't Hof, Yngve Villanger, Proper Interval Vertex Deletion, Algorithmica, DOI 10.1007/s00453-012-9661-3 |
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22.05.2013 Aleksandra Piktus |
Podstawy Informatyki On the Additive Constant of the k-Server Work Function Algorithm' by Yuval Emek, Pierre Fraigniaud, Amos Korman i Adi Rosen |
| We consider the Work Function Algorithm for the k-server problem (Chrobak andr Larmore, 1991; Koutsoupias and Papadimitriou, 1995). We show that if the Work Function Algorithm is c-competitive, then it is also strictly (2c)-competitive. As a consequence of (Koutsoupias and Papadimitriou, 1995) this also shows that the Work Function Algorithm is strictly (4k-2)-competitive. |
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16.05.2013 Grzegorz Stachowiak (UWr) |
Algorytmiczne Aspekty Kombinatoryki Counting and generating linear extensions |
| I begin with the problem of computing two numbers related to a partial order: the number of linear extensions e(P) and the parity difference d(P). This second numer is the difference between the number of linear extensions being odd and even permutations. The number d(P) was considered because the condition d(P)=0,1 is a necessary for the existence of of an algorithm generating linear extensions by transpositions. Both e(P) and d(P) are comparability invariants. Computing both of them is #P-hard. I will describe two simple algorithms to compute e(P) for special classes of posets. I will also formulate necessary and sufficient conditions for the possibility of generating permutations of a multiset by adjacent transpositions. |
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15.05.2013 Paweł Komosa |
Informatyka Teoretyczna Multicut viewed through the eyes of vertex cover |
| References:Jianer Chen, Jiahao Fany, Iyad A. Kanjz, Yang Liux, Fenghui Zhang, Multicut viewed through the eyes of vertex cover |
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15.05.2013 Dawid Pustułka |
Podstawy Informatyki An alternate proof of Statman's finite completeness theorem by B. Srivathsan, Igor Walukiewicz |
| Statman's finite completeness theorem says that for every pair of non-equivalent terms of simply-typed lambda-calculus there is a model that separates them. A direct method of constructing such a model is provided using a simple induction on the Böhm tree of the term. |
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09.05.2013 Piotr Wójcik |
Algorytmiczne Aspekty Kombinatoryki On algebraic invariants of geometric graphs; the Colin de Verdiere number. |
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08.05.2013 Sebastian Syta |
Informatyka Teoretyczna A Sublinear Time Algorithm for PageRank Computations |
| In a network, identifying all vertices whose PageRank is more than a given threshold value $\Delta$ is a basic problem that has arisen in Web and social network analyses. In this paper, we develop a nearly optimal, sublinear time, randomized algorithm for a close variant of this problem. When given a directed network \graph, a threshold value $\Delta$, and a positive constant $c>3$, with probability $1-o(1)$, our algorithm will return a subset $S\subseteq V$ with the property that $S$ contains all vertices of PageRank at least $\Delta$ and no vertex with PageRank less than $\Delta/c$. The running time of our algorithm is always $\tilde{O}(\frac{n}{\Delta})$. In addition, our algorithm can be efficiently implemented in various network access models including the Jump and Crawl query model recently studied by \cite{brautbar_kearns10}, making it suitable for dealing with large social and information networks. As part of our analysis, we show that any algorithm for solving this problem must have expected time complexity of ${\Omega}(\frac{n}{\Delta})$. Thus, our algorithm is optimal up to logarithmic factors. Our algorithm (for identifying vertices with significant PageRank) applies a multi-scale sampling scheme that uses a fast personalized PageRank estimator as its main subroutine. For that, we develop a new local randomized algorithm for approximating personalized PageRank which is more robust than the earlier ones developed by Jeh and Widom \cite{JehW03} and by Andersen, Chung, and Lang \cite{AndersenCL06}. References:Christian Borgs, Michael Brautbar, Jennifer Chayes1, and Shang-Hua Teng, A Sublinear Time Algorithm for PageRank Computations, |
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08.05.2013 Aneta Pawłowska |
Podstawy Informatyki TETRAVEX is NP-complete by Yasuhiko Takenaga and Toby Walsh |
| Tetravex is a widely played one person computer game in which you are given n^2 unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a n by n square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing Tetravex is computationally hard. More precisely, we prove that deciding if there is a tiling of the Tetravex board given n^2 unit tiles is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why Tetravex is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been show for: the offline version of Tetris, KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network), and shortest sliding puzzle problems. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate Tetravex problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions? |
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24.04.2013 Agnieszka Dymel |
Informatyka Teoretyczna A Simple 3-Edge-Connected Component Algorithm |
| A simple linear-time algorithm for finding all the 3-edge-connected components of an undirected graph is presented. The algorithm performs only one depth-first search over the given graph. Previously best known algorithms perform multiple depth-first searches in multiple phases. References:Yung H.Tsin, A Simple 3-Edge-Connected Component Algorithm, Theory Comput. Systems 40(2007), 125-142 |
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17.04.2013 Tomasz Kołodziejski |
Informatyka Teoretyczna 8/5 Approximation for TSP Paths |
| We prove the approximation ratio 8/5 for the metric s-t-Path-TSP problem, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple ``Best of Many'' version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s-t-tour out of those constructed from a family Fscr_{>0} of trees having a convex combination dominated by an optimal solution x^* of the fractional relaxation. They give the approximation guarantee sqrt{5}+1/2 for such an s-t--tour, which is the first improvement after the 5/3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x^*/2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in Fscr_{>0} into an s-t--path (or more generally, into a T-join) and its complement, which induces a decomposition of x^*. This decomposition can be refined and then efficiently used to complete x^*/2 without using linear programming or particular properties of T, but by adding to each cut deficient for x^*/2 an individually tailored explicitly given vector, inherent in the problem. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s-t--tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation. References:András Sebö, Eight-Fifth Approximation for TSP Paths, Integer Programming and Combinatorial Optimization, LNCS7801, 2013, pp 362-374 |
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17.04.2013 Monika Krupnik |
Podstawy Informatyki Inclusion problems for patterns with a bounded number of variables by Joachim Bremer, Dominik D. Freydenberger |
| We study the inclusion problems for pattern languages that are generated by patterns with a bounded number of variables. This continues the work by Freydenberger and Reidenbach (Information and Computation 208 (2010)) by showing that restricting the inclusion problem to significantly more restricted classes of patterns preserves undecidability, at least for comparatively large bounds. For smaller bounds, we prove the existence of classes of patterns with complicated inclusion relations, and an open inclusion problem, that are related to the Collatz Conjecture. In addition to this, we give the first proof of the undecidability of the inclusion problem for NE-pattern languages that, in contrast to previous proofs, does not rely on the inclusion problem for E-pattern languages, and proves the undecidability of the inclusion problem for NE-pattern languages over binary and ternary alphabets. |
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10.04.2013 Szymon Borak |
Informatyka Teoretyczna On dominating sets of maximal outerplanar graphs |
| A dominating set of a graph is a set S of vertices such that every vertex in the graph is either in S or is adjacent to a vertex in S. The domination number of a graph G, denoted gamma(G), is the minimum cardinality of a dominating set of G. We show that if G is an n-vertex maximal outerplanar graph, then gamma(G)≤(n+t)/4, where t is the number of vertices of degree 2 in G. We show that this bound is tight for all t≥2. Upper-bounds for gamma(G) are known for a few classes of graphs. References:C.N.Campos and Y.Wakabayashi, On dominating sets of maximal outerplanar graphs, Discrete Appl.Math. 161(2013), 330-335 |
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10.04.2013 Borg Łojasiewicz |
Podstawy Informatyki The state complexities of some basic operations on regular languages by Sheng Yu, Qingyu Zhuang and Kai Salomaa |
| We consider the state complexities of some basic operations on regular languages. We show that the number of states that is sufficient and necessary in the worst case for a deterministic finite automaton DFA) to accept the catenation of an m-state DFA language and an n-state DFA language is exactly m2^n - 2^{n-1} for m,n> 1. The result of 2^{n-1}+2^{n-2} states is obtained for the star of an n-state DFA language, n>1. State complexities for other basic operations and for regular languages over a one-letter alphabet are also studied. |
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03.04.2013 Aneta Pawłowska |
Informatyka Teoretyczna A Randomized O(log^2 k)-Competitive Algorithm for Metric Bipartite Matching |
| We consider the online metric matching problem in which we are given a metric space, k of whose points are designated as servers. Over time, up to k requests arrive at an arbitrary subset of points in the metric space, and each request must be matched to a server immediately upon arrival, subject to the constraint that at most one request is matched to any particular server. Matching decisions are irrevocable and the goal is to minimize the sum of distances between the requests and their matched servers. We give an O(log^2 k)-competitive randomized algorithm for the online metric matching problem. This improves upon the best known guarantee of O(log^3 k) on the competitive factor due to Meyerson, Nanavati and Poplawski (SODA'06). It is known that for this problem no deterministic algorithm can have a competitive better than 2k−1, and that no randomized algorithm can have a competitive ratio better than ln k. References:Nikhil Bansal, Niv Buchbinder, Anupam Gupta, Joseph (Seffi) Naor, A Randomized O(log^2 k)-Competitive Algorithm for Metric Bipartite Matching, Algorithmica, DOI 10.1007/s00453-012-9676-9 |
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03.04.2013 Michał Bejda |
Podstawy Informatyki Subshifts, Languages and Logic by Emmanuel Jeandel and Guillaume Theyssier |
| We study the Monadic Second Order (MSO) Hierarchy over infinite pictures, that is tilings. We give a characterization of existential MSO in terms of tilings and projections of tilings. Conversely, we characterise logic fragments corresponding to various classes of infinite pictures (subshifts of finite type, sofic subshifts). |
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28.03.2013 Robert Obryk |
Algorytmiczne Aspekty Kombinatoryki Topological structure of asynchronous computability |
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27.03.2013 Michał Sapalski |
Informatyka Teoretyczna A Lower Bound of 1+ϕ for Truthful Scheduling Mechanisms |
| We study the mechanism design version of the unrelated machines scheduling problem, which is at the core of Algorithmic Game Theory and was first proposed and studied in a seminal paper of Nisan and Ronen. We give an improved lower bound of 1+ϕ≈2.618 on the pproximation ratio of deterministic truthful mechanisms for the makespan. The proof is based on a recursive preprocessing argument which yields a strictly increasing series of new lower bounds for each fixed number of machines n≥4. References:Elias Koutsoupias, Angelina Vidali, A Lower Bound of 1+ϕ for Truthful Scheduling Mechanisms, Algorithmica, DOI 10.1007/s00453-012-9634-6 |
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27.03.2013 Jacek Szmigiel |
Podstawy Informatyki Bad news on decision problems for patterns by Dominik D. Freydenberger, Daniel Reidenbach |
We study the inclusion problem for pattern languages, which—due to Jiang et al. [T. Jiang, A. Salomaa, K. Salomaa, S. Yu, Decision problems for patterns, Journal of Computer and System Sciences 50 (1995) 53–63]— is known to be undecidable. More precisely, Jiang et al. demonstrate that there is no effective procedure deciding the inclusion for the class of all pattern languages over all alphabets. Most applications of pattern languages, however, consider classes over fixed alphabets, and therefore it is practically more relevant to ask for the existence of alphabet-specific decision procedures. Our first main result states that, for all but very particular cases, this version of the inclusion problem is also undecidable. The second main part of our paper disproves the prevalent conjecture on the inclusion of so-called similar E-pattern languages, and it explains the devastating consequences of this result for the intensive previous research on the most prominent open decision problem for pattern languages, namely the equivalence problem for general E-pattern languages. |
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21.03.2013 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki Nonrepetitive colourings of planar graphs with O(log n) colours |
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13.03.2013 Marek Markiewicz |
Podstawy Informatyki Topology on words |
| During the talk we will explore two types of topologies on the set of all infinite words over a finite alphabet with at least two different letters: the Cantor topology and its relative version U^\delta topology for an arbitrary languge U of finite words. We will describe close and open sets in both topologies and how they relate to each other. We will also explore the definitions of Chaitin and Martin-Löf random sequences and will prove their equivalence. Finally we will show that the set of Martin-Löf random sequences is co-nowhere dense in U^\delta topology for a special U. The talk is based on three papers: Topological characterization of random sequences by C. Calude, S. Marcus, L. Steiger; Weighted Finite Automata and Mertics in Cantor Space by L. Steiger and Exploring Randomness by G. Chaitin. |
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07.03.2013 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki The List Coloring Conjecture for planar graphs |
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28.02.2013 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Perspectives in the Online Ramsey Theory |
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24.01.2013 Maciej Kalkowski (UAM) |
Algorytmiczne Aspekty Kombinatoryki Irregularity strength in distributed model |
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23.01.2013 Michał Feret |
Informatyka Teoretyczna Relative Convex Hulls in Semi-Dynamic Arrangements |
| We present a data structure for maintaining the geodesic hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. For m barriers and n sites, our data structure has O((m+n)log(n)) size. It supports a mixed sequence of O(m) barrier insertions and O(n) site deletions in O((m+n)polylog(mn)) total time, and answers analogues of standard convex hull queries in O(polylog(mn)) time. Our data structure supports a generalization of the sweep line technique, in which the sweep wavefront is a simple closed polygonal curve, and it sweeps a set of n points in the plane by simple moves. We reduce the total time of supporting m online moves of a polygonal wavefront sweep algorithm from the naïve O(m√n polylog(n)) to O((m+n)polylog(mn)). References:Mashhood Ishaque, Csaba D. Tóth, Relative Convex Hulls in Semi-Dynamic Arrangements, Algorithmica DOI 10.1007/s00453-012-9679-6 |
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23.01.2013 Adam Polak |
Podstawy Informatyki On the Complexity of the Equivalence Problem for Probabilistic Automata by Stefan Kiefer, Andrzej S. Murawski, Jo¨el Ouaknine, Bj¨orn Wachter1, and James Worrell |
| Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero. |
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16.01.2013 Jacek Szmigiel |
Informatyka Teoretyczna An Optimal Lower Bound for Buffer Management in Multi-Queue Switches |
| In the online packet buffering problem (also known as the unweighted FIFO variant of buffer management), we focus on a single network packet switching device with several input ports and one output port. This device forwards unit-size, unit-value packets from input ports to the output port. Buffers attached to input ports may accumulate incoming packets for later transmission; if they cannot accommodate all incoming packets, their excess is lost. A packet buffering algorithm has to choose from which buffers to transmit packets in order to minimize the number of lost packets and thus maximize the throughput. We present a tight lower bound of e/(e−1) ≈ 1.582 on the competitive ratio of the throughput maximization, which holds even for fractional or randomized algorithms. This improves the previously best known lower bound of 1.4659 and matches the performance of the algorithm RANDOM SCHEDULE. Our result contradicts the claimed performance of the algorithm RANDOM PERMUTATION; we point out a flaw in its original analysis. References:Marcin Bieńkowski, An Optimal Lower Bound for Buffer Management in Multi-Queue Switches, Algorithmica DOI 10.1007/s00453-012-9677-8 |
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16.01.2013 Paweł Wanat |
Podstawy Informatyki The Local Lemma is Tight for SAT by H. Gebauer |
| We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))2^k/k clauses. The lopsided Local Lemma shows that our result is asymptotically best possible: every k-CNF formula where every variable appears in at most 2^(k+1)/e(k+1) 1 clauses is satisfiable. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YESinstance it becomes NP-hard just by allowing each variable to occur in one more clause. The asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = (1/e + o(1))2^k. The constructed formulas are all in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables and thus they resolve these asymptotic questions within that class as well. |
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09.01.2013 Jakub Adamek |
Informatyka Teoretyczna A Distributed O(1)-Approximation Algorithm for the Uniform Facility Location Problem |
| We investigate a metric facility location problem in a distributed setting. In this problem, we assume that each point is a client as well as a potential location for a facility and that the opening costs for the facilities and the demands of the clients are uniform. The goal is to open a subset of the input points as facilities such that the accumulated cost for the whole point set, consisting of the opening costs for the facilities and the connection costs for the clients, is minimized. We present a randomized distributed algorithm that computes in expectation an O(1)-approximate solution to the metric facility location problem described above. Our algorithm works in a synchronous message passing model, where each point is an autonomous computational entity that has its own local memory and that communicates with the other entities by message passing.We assume that each entity knows the distance to all the other entities, but does not know any of the other pairwise distances. Our algorithm uses three rounds of all-to-all communication with message sizes bounded to O(log(n)) bits, where n is the number of input points. We extend our distributed algorithm to constant powers of metric spaces. For a metric exponent l≥1, we obtain a randomized O(1)-approximation algorithm that uses three rounds of all-to-all communication with message sizes bounded to O(log(n)) bits. References:Joachim Gehweiler, Christiane Lammersen, Christian Sohler, A Distributed O(1)-Approximation Algorithm for the Uniform Facility Location Problem, Algorithmica DOI 10.1007/s00453-012-9690-y |
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09.01.2013 Andrzej Dorobisz |
Podstawy Informatyki Functions definable by numerical set-expressions by IAN PRATT-HARTMANN and IVO DÜNTSCH |
| A numerical set-expression is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of additive circuits. If they are confined to the usual Boolean operations together with the result of lifting addition and multiplication to the level of sets, we speak of arithmetic circuits. In this article, we investigate the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations. |
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03.01.2013 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Old and new challenges in Thue theory |
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20.12.2012 Bartosz Walczak |
Algorytmiczne Aspekty Kombinatoryki Optimal tree-grabbing |
| Alice and Bob share an unrooted tree with non-negative weights assigned to the vertices, and play a game on it. In the first move, Alice cuts a leaf of the tree and scores its weight. Then, Bob and Alice alternate turns, in each move cutting a leaf of the remaining tree and adding its weight to their own score. Their goal in the game is to maximize their own final score. This game has been introduced in a joint paper with Micek [1], where we proved that Alice can guarantee herself at least 1/4 of the total weight of the tree, and conjectured that actually she can do at least 1/2. This conjecture has been proved by Seacrest and Seacrest [2]. Now, an intriguing open problem is to devise a polynomial-time algorithm computing an optimal move at any position of the game. In this talk, I will share my thoughts of what such an algorithm may look like, and ask the audience for a proof of correctness or a counterexample :) References:[1] Piotr Micek, Bartosz Walczak, A graph-grabbing game, Combinatorics, Probability and Computing 20 (2011), 623-629 [2] Deborah E. Seacrest, Tyler Seacrest, Grabbing the gold, Discrete Mathematics 312 (2012), 1804-1806 |
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19.12.2012 Radoslaw Smyrek |
Informatyka Teoretyczna Recognizing d-Interval Graphs and d-Track Interval Graphs |
| A d-interval is the union of d disjoint intervals on the real line. A d-track interval is the union of d disjoint intervals on d disjoint parallel lines called tracks, one interval on each track. As generalizations of the ubiquitous interval graphs, d-interval graphs and d-track interval graphs have wide applications, traditionally to scheduling and resource allocation, and more recently to bioinformatics. In this paper, we prove that recognizing d-track interval graphs is NP-complete for any constant d≥2. This confirms a conjecture of Gyárfás and West in 1995. Previously only the complexity of the case d=2 was known. Our proof in fact implies that several restricted variants of this graph recognition problem, i.e., recognizing balanced d-track interval graphs, unit d-track interval graphs, and (2,..., 2) d-track interval graphs, are all NP-complete. This partially answers another question recently raised by Gambette and Vialette. We also prove that recognizing depth-two 2-track interval graphs is NP-complete, even for the unit case. In sharp contrast, we present a simple linear-time algorithm for recognizing depth-two unit d-interval graphs. These and other results of ours give partial answers to a question of West and Shmoys in 1984 and a similar question of Gyárfás and West in 1995. Finally, we give the first bounds on the track number and the unit track number of a graph in terms of the number of vertices, the number of edges, and the maximum degree, and link the two numbers to the classical concepts of arboricity. References:Minghui Jiang: Recognizing d-Interval Graphs and d-Track Interval Graphs, Algorithmica DOI 10.1007/s00453-012-9651-5 |
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19.12.2012 Przemysław Jedynak |
Podstawy Informatyki SUBWORD OCCURRENCES, PARIKH MATRICES AND LYNDON IMAGES by ARTO SALOMAA and SHENG YU |
| We investigate the number of occurrences of a word u as a (scattered) subword of a word w. The notion of a Parikh matrix, recently introduced, is a basic tool in this investigation. In general, several words are associated with a Parikh matrix. The ambiguity can be resolved by associating a unique word called the Lyndon image to each Parikh matrix. In this paper we will investigate properties of Lyndon images and the corresponding questions of ambiguity. We give an exhaustive characterization in the case of a binary alphabet. Our main results in the general case deal with the comparison of unambiguous words and Lyndon images, algorithms for constructing Lyndon images, as well as classes of words with the same Parikh matrix, obtained by circular variance. |
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13.12.2012 Michał Sapalski |
Algorytmiczne Aspekty Kombinatoryki Finding minimum-weight (undirected) spanning tree for process networks |
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12.12.2012 Jarosław Bielenin |
Informatyka Teoretyczna Graph Balancing: A Special Case of Scheduling Unrelated Parallel Machines |
| We design a 1.75-approximation algorithm for a special case of scheduling parallel machines to minimize the makespan, namely the case where each job can be assigned to at most two machines, with the same processing time on either machine. (This is a special case of so-called restricted assignment, where the set of eligible machines can be arbitrary for each job.) This is the first improvement of the approximation ratio 2 of Lenstra, Shmoys, and Tardos (Math. Program. 46:259–271, 1990), to a smaller constant for any special case with unbounded number of machines and unbounded processing times.We conclude by showing integrality gaps of several relaxations of related problems. References:Tomáš Ebenlendr, Marek Krˇcál, Jiˇrí Sgall: Graph Balancing: A Special Case of Scheduling Unrelated Parallel Machines, Algorithmica DOI 10.1007/s00453-012-9668-9 |
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12.12.2012 Maciej Bendkowski |
Podstawy Informatyki NONDETERMINISTIC FINITE AUTOMATA RECENT RESULTS ON THE DESCRIPTIONAL AND COMPUTATIONAL COMPLEXITY by MARKUS HOLZER and MARTIN KUTRIB |
continuation |
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05.12.2012 Michał Masłowski |
Informatyka Teoretyczna On the Exact Complexity of Evaluating Quantified k-CNF |
| We relate the exponential complexities 2^{s(k)n} of k-SAT and the exponential complexity 2^{s(EVAL(Π_2 3-CNF))n} of EVAL(Π_2 3-CNF) (the problem of evaluating quantified formulas of the form ∀x∃yF(x,y) where F is a 3-CNF in x-variables and y-variables) and show that s(∞) (the limit of s(k) as k→∞) is at most s(EVAL(Π_2 3-CNF)). Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for EVAL(Π_2 3-CNF) running in time 2^{cn} with c<1. On the other hand, a nontrivial exponential-time algorithm for EVAL(Π_2 3-CNF) would provide a k-SAT solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem EVAL(Π_2 3-CNF) have nontrivial algorithms, and provide strong evidence that the hardest cases of EVAL(Π_2 3-CNF) must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n−o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable k-CNFs and the application of the Sparsification lemma. References:Chris Calabro, Russell Impagliazzo, Ramamohan Paturi, On the Exact Complexity of Evaluating Quantified k-CNF, Algorithmica DOI 10.1007/s00453-012-9648-0 |
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05.12.2012 Maciej Bendkowski |
Podstawy Informatyki NONDETERMINISTIC FINITE AUTOMATA RECENT RESULTS ON THE DESCRIPTIONAL AND COMPUTATIONAL COMPLEXITY by MARKUS HOLZER and MARTIN KUTRIB |
Nondeterministic finite automata (NFAs) were introduced in [68], where their equivalence to deterministic finite automata was shown. Over the last 50 years, a vast literature documenting the importance of finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss recent developments relevant to NFAs related problems like, for example, (i) simulation of and by several types of finite automata, (ii) minimization and approximation, (iii) size estimation of minimal NFAs, and (iv) state complexity of language operations. We thus come across descriptional and computational complexity issues of nondeterministic finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved. |
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28.11.2012 Agnieszka Łupińska |
Informatyka Teoretyczna Speed Scaling on Parallel Processors |
| In this paper we investigate dynamic speed scaling, a technique to reduce energy consumption in variable-speed microprocessors. While prior research has focused mostly on single processor environments, in this paper we investigate multiprocessor settings. We study the basic problem of scheduling a set of jobs, each specified by a release date, a deadline and a processing volume, on variable-speed processors so as to minimize the total energy consumption. We first settle the problem complexity if unit size jobs have to be scheduled. More specifically, we devise a polynomial time algorithm for jobs with agreeable deadlines and prove NP-hardness results if jobs have arbitrary deadlines. For the latter setting we also develop a polynomial time algorithm achieving a constant factor approximation guarantee. Additionally, we study problem settings where jobs have arbitrary processing requirements and, again, develop constant factor approximation algorithms. We finally transform our offline algorithms into constant competitive online strategies. References:Susanne Albers, Fabian Müller, Swen Schmelzer, Speed Scaling on Parallel Processors, Algorithmica DOI 10.1007/s00453-012-9678-7 |
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28.11.2012 Maciej Gawron |
Podstawy Informatyki Hilbert's tenth problem |
| The question of finding an algorithm to determine whether a given Diophantine equation has an integer solution, was one of the famous Hilbert's problems, posed in 1900. It was finally answered (negatively) by Yuri Matiyasevich in 1970. We will show the proof of this fact. We will introduce the notion of Diophantine sets, relations and functions. We will prove that Diophantine sets are exactly computably enumerable sets. We will show that there exists an universal Diophantine equation, then using standard diagonal method we will prove that Hilbert's tenth problem is undecidable. |
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21.11.2012 Gabriel Fortin |
Informatyka Teoretyczna 3-Colouring AT-Free Graphs in Polynomial Time |
| Determining the complexity of the colouring problem on AT-free graphs is one of long-standing open problems in algorithmic graph theory. One of the reasons behind this is that AT-free graphs are not necessarily perfect unlike many popular subclasses of AT-free graphs such as interval graphs or co-comparability graphs. In this paper, we resolve the smallest open case of this problem, and present the first polynomial time algorithm for the 3-colouring problem on AT-free graphs. References:Juraj Stacho, 3-Colouring AT-Free Graphs in Polynomial Time , Algorithmica (2012) 64:384–399 |
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21.11.2012 Arkadiusz Olek |
Podstawy Informatyki Verifiable secret sharing in a total of three rounds by Shashank Agrawal |
| Verifiable secret sharing (VSS) is an important building block in the design of secure multiparty protocols, when some of the parties are under the control of a malicious adversary. Henceforth, its round complexity has been the subject of intense study. The best known unconditionally secure protocol takes 3 rounds in sharing phase, which is known to be optimal, and 1 round in reconstruction. Recently, by introducing a negligible probability of error in the definition of VSS, Patra et al. [CRYPTO 2009] have designed a novel protocol which takes only 2 rounds in sharing phase. However, the drawback of their protocol is that it takes 2 rounds in reconstruction as well. Hence, the total number of rounds required for VSS remains the same. In this paper, we present a VSS protocol which takes a total of 3 rounds only—2 rounds in sharing and 1 round in reconstruction. |
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14.11.2012 Łukasz Janiszewski |
Informatyka Teoretyczna The Complexity of the Empire Colouring Problem |
| We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in Q. J. Pure Appl. Math. 24:332–338, 1890) for maps containing empires formed by exactly r>1 countries each. We prove that the problem can be solved in polynomial time using s colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than s/r. However, if s≥3, the problem is NP-hard even if the graph is a forests of paths of arbitrary lengths (for any r≥2, provided s<2r−\sqrt{2r+1/4}+3/2. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any r≥2, if 3≤s≤2r−1 (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if s<7 for r=2, and s<6r−3, for r≥3. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than 6r−3 colours graphs of thickness r≥3. References:Andrew R.A. McGrae, Michele Zito, The Complexity of the Empire Colouring Problem, Algorithmica DOI 10.1007/s00453-012-9680-0 |
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14.11.2012 Michał Marczyk |
Podstawy Informatyki Unification type of bounded distributive lattices |
| We will present S. Ghilardi's proof of his unification type theorem for bounded distributive lattices. The focus will be on the main result; a high-level overview of the underlying methodology will be presented without detailed proofs of the individual results (which have been discussed in this seminar at an earlier date). The theorem as well as the methodology employed in establishing it have been presented in S. Ghilardi, "Unification through Projectivity", Journal of Logic and Computation (1997) 7. |
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08.11.2012 Robert Obryk |
Algorytmiczne Aspekty Kombinatoryki A near-optimal cardinality estimation algorithm |
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07.11.2012 Maciej Bendkowski |
Informatyka Teoretyczna Sex-Equal Stable Matchings: Complexity and Exact Algorithms |
| We explore the complexity and exact computation of a variant of the classical stable marriage problem in which we seek matchings that are not only stable, but are also "fair" in a formal sense. In particular, we study the sex-equal stable marriage problem (SESM), in which, roughly speaking, we wish to find a stable matching with the property that the men's happiness is as close as possible to the women's happiness. This problem is known to be strongly NP-hard References:Eric McDermid, Robert W. Irving, Sex-Equal Stable Matchings: Complexity and Exact Algorithms, Algorithmica, DOI 10.1007/s00453-012-9672-0 |
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07.11.2012 Katarzyna Grygiel |
Podstawy Informatyki Finite lattices and their weighted double skeletons |
| In 1973 Christian Herrmann introduced the notion of the skeleton of a finite lattice. The skeleton of a lattice, however, does not suffice to reconstruct the initial lattice uniquely. Even worse, it turns out that every finite lattice is the skeleton of infinitely many non-isomorphic distributive lattices. At this point a natural question arises whether one can stuff the skeleton with some additional data in order to restore the original lattice. During the talk I will focus on the so-called weighted double skeletons. These objects, not being lattices themselves, turn out to fully characterize a particular kind of lattices. |
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31.10.2012 Tomasz JurkiewiczMax Planck Institute for Informatics, Saarbrücken |
Informatyka Teoretyczna The Cost of Address Translation |
| Modern computers are not random access machines (RAMs). They have a memory hierarchy, multiple cores, and virtual memory. We address the problem of the computational cost of address translation in virtual memory. Starting point for our work is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the EM model (external memory model) does not correctly predict growth rates of actual running times. We propose the VAT model (virtual address translation) to account for the cost of address translations and analyze the algorithms mentioned above and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cache-oblivious algorithms. References:Tomasz Jurkiewicz and Kurt Mehlhorn, The Cost of Address Translation, ALENEX, January 2013. |
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31.10.2012 Michał Sapalski |
Podstawy Informatyki The non-uniform Bounded Degree Minimum Diameter Spanning Tree problem with an application in P2P networking by Jakarin Chawachat, Jittat Fakcharoenphol, Wattana Jindaluang |
This paper considers the Bounded Degree Minimum Diameter Spanning Tree problem (BDST problem) with non-uniform degree bounds. In this problem, we are given a metric length function over a set V of n nodes and a degree bound Bv for each v ∈ V, and want to find a spanning tree with minimum diameter such that each node v has degree at most Bv . We present a simple extension of an O(logn)-approximation algorithm for this problem with uniform degree bounds of Könemann, Levin, and Sinha [J. Könemann, A. Levin, A. Sinha, Approximating the degree-bounded minimum diameter spanning tree problem, in: Algorithmica, Springer, 2003, pp. 109–121] to work with nonuniform degree bounds. We also show that this problem has an application in the peerto-peer content distribution. More specifically, the Minimum Delay Mesh problem (MDM problem) introduced by Ren, Li and Chan [D. Ren, Y.-T. Li, S.-H. Chan, On reducing mesh delay for peer-to-peer live streaming, in: INFOCOM, 2008, pp. 1058–1066] under a natural assumption can be reduced to this non-uniform case of the BDST problem. |
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25.10.2012 Andrzej Pezarski |
Algorytmiczne Aspekty Kombinatoryki On-line clique covering of interval graphs II |
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24.10.2012 Adam Polak |
Informatyka Teoretyczna Algorithms for Placing Monitors in a Flow Network |
| References:Francis Chin, Marek Chrobak, Li Yan, Algorithms for Placing Monitors in a Flow Network |
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24.10.2012 Agnieszka Łupińska |
Podstawy Informatyki On building minimal automaton for subset matching queries by Kimmo Fredriksson |
We address the problem of building an index for a set D of n strings, where each string location is a subset of some finite integer alphabet of size σ, so that we can answer efficiently if a given simple query string (where each string location is a single symbol) p occurs in the set. That is, we need to efficiently find a string d ∈ D such that p[i] ∈ d[i] for every i. We show how to build such index in O(nlogσ/(σ) log(n)) average time, where is the average size of the subsets. Our methods have applications e.g. in computational biology (haplotype inference) and music information retrieval. |
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18.10.2012 Andrzej Pezarski |
Algorytmiczne Aspekty Kombinatoryki On-line clique covering of interval graphs |
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17.10.2012 Lech Duraj |
Informatyka Teoretyczna A linear algorithm for 3-letter LCWIS problem |
| The problem of finding longest weakly increasing common subsequence (LCWIS) of two sequences is a variant of popular longest common subsequence (LCS) problem. While there are no known methods to find LCS in truly sub-quadratic time, there are faster algorithms to compute LCWIS if the alphabet size is small enough. We present a linear-time algorithm finding LCWIS over 3-letter alphabet. Up to now, the fastest known algorithm was O(min{m + n log n, m log log m}). |
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17.10.2012 Michał Masłowski |
Podstawy Informatyki Regular patterns, regular languages and context-free languages by Sanjay Jain, Yuh Shin Ong, Frank Stephan |
| In this paper we consider two questions. First we consider whether every pattern language which is regular can be generated by a regular pattern. We show that this is indeed the case for extended (erasing) pattern languages if alphabet size is at least four. In all other cases, we show that there are patterns generating a regular language which cannot be generated by a regular pattern. Next we consider whether there are pattern languages which are context-free but not regular. We show that, for alphabet size 2 and 3, there are both erasing and non-erasing pattern languages which are context-free but not regular. On the other hand, for alphabet size at least 4, every erasing pattern language which is context-free is also regular. It is open at present whether there exist non-erasing pattern languages which are context-free but not regular for alphabet size at least 4. |
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11.10.2012 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Coloring problems for graphs and matroids |
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10.10.2012 Ariel GabizonTechnion |
Informatyka Teoretyczna Invertible Zero-Error Dispersers and Defective Memory with Stuck-At Errors |
| Kuznetsov and Tsybakov considered the problem of storing information in a memory where a certain p-fraction of the n cells are `stuck' at certain values. The person writing in the memory - the `encoder'- knows which cells are stuck, and to what values. The person who will read the memory later - the `decoder' is required to retrieve the message encoded *without* the information about which cells are stuck. Kuznetsov and Tsybakov showed there are schemes where a message of length (1-p-o(1))*n can be encoded. We give the first such explicit schemes. Our schemes follow from a construction of an object called an `invertible zero-error disperser'. Joint work with Ronen Shaltiel.
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10.10.2012 Piotr Wójcik |
Podstawy Informatyki Some results about decidability of sets of tautologies in first order languages. |
| The fundamental question whether a set of first order tautologies is decidable was answered (negatively) by Church in 1936. By restricting the classes of considered sentences (e.g. by reducing the number of function symbols or the arity of predicates), we can produce some positive results. After exploring well-known languages, we will move on to study more complex systems, like first order monadic temporal logic. There, even without function symbols and equality, the set of tautologies is not recursively enumerable. |
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03.10.2012 Mateusz Kostanek |
Podstawy Informatyki Reconciliation of elementary order and metric fixpoint theorems |
| We prove two new fixed point theorems for generalized metric spaces and show that various fundamental fixed point principles, including: Banach Contraction Principle, Caristi fixed point theorem for metric spaces, Knaster-Tarski fixed point theorem for complete lattices, and the Bourbaki-Witt fixed point theorem for directed-complete orders, follow as corollaries of our results. |
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20.06.2012 Szymon Borak |
Informatyka Teoretyczna Monadic Second Order Logic on Graphs with Local Cardinality Constraints |
| We show that all problems of the following form can be solved in polynomial time for graphs of bounded treewidth: Given a graph G and for each vertex v of G a set α(v) of non-negative integers. Is there a set S of vertices or edges of G such that S satisfies a fixed property expressible in monadic second order logic, and for each vertex v of G the number of vertices/edges in S adjacent/incident with v belongs to the set α(v)? A wide range of problems can be formulated in this way, for example Lovasz's General Factor Problem. References:Stefan Szeider, Monadic Second Order Logic on Graphs with Local Cardinality Constraints, LNCS 5162, pp. 601–612, 2008. |
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14.06.2012 Michał Kukieła |
Algorytmiczne Aspekty Kombinatoryki Algebraic topology applied to evasiveness of graph properties |
| The evasiveness conjecture (also known as the Aanderaa-Karp-Rosenberg conjecture) states that any non-trivial monotone property P of graphs on a fixed set of n vertices (i.e. a property closed under removing edges) is evasive, which means that given an unknown graph G on the n vertices and allowed to ask whether a given edge belongs to G, we need in the worst case to ask about all possible n(n-1)/2 edges in order to determine whether G has the property P or not. In other words, P has decision tree complexity n(n-1)/2. I will discuss the classical paper of Jeff Kahn, Michael Saks and Dean Sturtevant that applies techniques of algebraic topology to this conjecture, proving it in the case when n is a prime power. If there is enough time left I shall give a short survey of some recent results in this area. |
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13.06.2012 Marek Markiewicz |
Informatyka Teoretyczna Sharp Separation and Applications to Exact and Parameterized Algorithms |
| Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications. References:Fedor V. Fomin, Fabrizio Grandoni, Daniel Lokshtanov and Saket Saurabh, Sharp Separation and Applications to Exact and Parameterized Algorithms, Algorithmica, DOI 10.1007/s00453-011-9555-9 |
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13.06.2012 Michał Marczyk |
Podstawy Informatyki Persistent data structures |
Persistent data structures, that is data structures which are immutable and support efficient creation of slightly modified copies with no change to the complexity guarantees of the basic operations (both on the copy and on the original) are of key importance for the performance of programs written in the functional paradigm. We will examine in some detail a single example, namely a hash table offering logarithmic complexity of the basic operations with very good constants. The data structure in question is based on the mutable Hash Array Mapped Trie described in Phil Bagwell's paper "Ideal Hash Trees" [1] (see also Phil Bagwell, "Fast And Space Efficient Trie Searches" [2]). The persistent version was pioneered by Rich Hickey and is used in the Clojure programming language [3], [4], [5] ([4] -- the Java implementation used in Clojure; [5] -- the ClojureScript port of [4] used in ClojureScript). [1] http://lampwww.epfl.ch/papers/idealhashtrees.pdf [2] http://lampwww.epfl.ch/papers/triesearches.pdf.gz [3] http://clojure.org/ |
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06.06.2012 30.05.2012,Andrzej Pezarski |
Informatyka Teoretyczna On-line clique covering of unit interval graphs |
| We consider an on-line version of the minimal clique covering problem. We focus on a class of unit interval graphs and their different representations. It is known that all greedy algorithms solving this roblems use at least two times more cliques in the worst scenario than it is necessary in the optimal off-line solution. We introduce non-greedy approach, which leads us to construction of new better algorithms. We start with connected graphs presented in a connected way with their proper interval representations. For this case we show an algorithm using at worst 8/5 times more cliques than it is needed. Later, we generalize this solution to the case of non-connected graphs. This time, we obtain an algorithm using at worst 13/8 times more cliques than it is necessary. We also generalize both algorithms to work without interval representation. Finally, we move towards unit interval representation and present an algorithm using at most 8/5 times more cliques than needed. The performance of the algorithms is the best possible.
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06.06.2012 Michał Handzlik |
Podstawy Informatyki Asymetric grammars |
| Asymetric gramars are natural generalization of leftist grammars. Leftist grammars were introduced by Motwani as a tool useful to study accessibility problems in some protection systems. Since then, some interesting language-theoretical research has been carried out in this field. For example, the membership problem for leftist grammar is decidable. We propose a natural way to generalize the notion od leftist grammar. We study how this generalization affects the location of languages generated by those grammars within the Chomsky hierarchy. The main result states that membership problem for asymetric grammars is undecidable. |
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31.05.2012 Karol Kosiński |
Algorytmiczne Aspekty Kombinatoryki A regularity lemma and twins in words |
| For a word S, let f(S) be the largest integer m such that there are two disjoints identical (scattered) subwords of length m. Let f(n,A) = min{f(S) : S is of length n, over alphabet A}. Here, it is shown that 2f(n,{0,1}) = n − o(n) using the regularity lemma for words. I.e., any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length o(n). A similar result is proven for k identical subwords of a word over an alphabet with at most k letters. |
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30.05.2012 Marek Markiewicz |
Podstawy Informatyki A new class of hyper-bent Boolean functions in binomial forms by Baocheng Wang, Chunming Tang, Yanfeng Qi, Yixian Yang and Maozhi Xu |
| Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals 2^{n−1} ± 2^{n/2 −1}, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent unctions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. |
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24.05.2012 Piotr Faliszewski (AGH) |
Algorytmiczne Aspekty Kombinatoryki Clone Structures in Voters' Preferences |
| In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is a family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show their hierarchical structure, and analyze clone structures in single-peaked and single-crossing elections. We give a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to become single-peaked, and we show that this problem is NP-hard for single-crossing elections. Joint work with Edith Elkind and Arkadii Slinko, to be presented at 13th ACM Conference on Electronic Commerce. The paper is available at: http://home.agh.edu.pl/~faliszew/decloning.pdf |
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23.05.2012 Maciej Wawro |
Informatyka Teoretyczna Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming |
| The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U+V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint. We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by |V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes W[1]-hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic instances, each of which can be solved in polynomial time by dynamic programming. References:Gregory Gutin, Eun Jung Kim, Arezou Soleimanfallah, Stefan Szeider and Anders Yeo, Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming, Algorithmica, DOI 10.1007/s00453-011-9548-8 |
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23.05.2012 Maciej Bendkowski |
Podstawy Informatyki On the expressive power of schemes by Dowek G, Jiang Y |
| We present a calculus, called the scheme-calculus, that permits to express natural deduction proofs in various theories. Unlike lambda-calculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for the hypotheses. We show that despite its non-determinism, some typed scheme-calculi have the same expressivity as the corresponding typed lambda-calculi. |
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17.05.2012 Paweł Dłotko |
Algorytmiczne Aspekty Kombinatoryki Forman's discrete Morse theory + applications |
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16.05.2012 Kasper Kopeć |
Informatyka Teoretyczna Minimum Weight Cycles and Triangles: Equivalences and Algorithms |
| We consider the fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph. In particular, we show that the minimum weight cycle problem in an undirected n-node graph with edge weights in {1,...,M} or in a directed n-node graph with edge weights in {-M,..., M} and no negative cycles can be efficiently reduced to finding a minimum weight triangle in an Theta(n)-node undirected graph with weights in {1,...,O(M)}. Roughly speaking, our reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be "encoded" using only three edges within roughly the same weight interval! This resolves a longstanding open problem posed by Itai and Rodeh [SIAM J. Computing 1978 and STOC'77]. A direct consequence of our efficient reductions are O (Mn^{omega})-time algorithms using fast matrix multiplication (FMM) for finding a minimum weight cycle in both undirected graphs with integral weights from the interval [1,M] and directed graphs with integral weights from the interval [-M,M]. The latter seems to reveal a strong separation between the all pairs shortest paths (APSP) problem and the minimum weight cycle problem in directed graphs as the fastest known APSP algorithm has a running time of O(M^{0.681}n^{2.575}) by Zwick [J. ACM 2002]. > In contrast, when only combinatorial algorithms are allowed (that is, without FMM) the only known solution to minimum weight cycle is by computing APSP. Interestingly, any separation between the two problems in this case would be an amazing breakthrough as by a recent paper by Vassilevska W. and Williams [FOCS'10], any O(n^{3-eps})-time algorithm (eps>0) for minimum weight cycle immediately implies a O(n^{3-delta})-time algorithm (delta>0) for APSP. References:Liam Roditty and Virginia Vassilevska Williams, Minimum Weight Cycles and Triangles: Equivalences and Algorithms, http://arxiv.org/abs/1104.2882v1 |
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10.05.2012 Andrzej Kisielewicz, Krzysztof Przesławski (UZ) |
Algorytmiczne Aspekty Kombinatoryki On the structure of tilings of Euclidean space by unit cubes -- around (disproved) Keller's conjecture |
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09.05.2012 Maciej Gawron |
Informatyka Teoretyczna An exact algorithm for the Maximum Leaf Spanning Tree problem |
| Given an undirected graph with n vertices, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4^k poly(n)) using a simple branching algorithm introduced by a subset of the authors (Kneis et al. 2008). Daligault et al. (2010) improved the branching and obtained a running time of O(3.72^k poly(n)). In this paper, we study the problem from an exponential time viewpoint, where it is equivalent to the Connected Dominating Set problem. Here, Fomin, Grandoni, and Kratsch showed how to break the Ω(2^n) barrier and proposed an O(1.9407^n)-time algorithm (Fomin et al. 2008). Based on some useful properties of Kneis et al. (2008) and Daligault et al. (2010), we present a branching algorithm whose running time of O(1.8966^n) has been analyzed using the Measure-and-Conquer technique. Finally, we provide a lower bound of Ω(1.4422^n) for the worst case running time of our algorithm. References:Henning Fernau, Joachim Kneis, Dieter Kratsch, Alexander Langer, Mathieu Liedloff, Daniel Raible, Peter Rossmanith, An exact algorithm for the Maximum Leaf Spanning Tree problem, Theoretical Computer Science 412(2011) 6290–6302 |
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09.05.2012 Piotr Zaborski |
Podstawy Informatyki APPROXIMATION ALGORITHMS FOR THE EUCLIDEAN TRAVELING SALESMAN PROBLEM WITH DISCRETE AND CONTINUOUS NEIGHBORHOODS by KHALED ELBASSIONI, ALEKSEI V. FISHKIN and RENE SITTERS |
In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to nd a tour of minimum length which visits at least one point in each region. We give (i) an O(\alpha)-approximation algorithm for the case when the regions are disjoint and -fat, with possibly varying size; (ii) an O(\alpha^3)- approximation algorithm for intersecting -fat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(logn)-approximation algorithm for continuous non-fat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low running-time complexities. |
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26.04.2012 Gwenaël Joret |
Algorytmiczne Aspekty Kombinatoryki Excluded Forest Minors and the Erdos-Posa Property |
| A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the so-called Erdos-Posa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, either G has k vertex-disjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with |X| \leq f(k) such that G - X has no H-minor. While the best function f currently known is super-exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour, and Thomas on the pathwidth of graphs with an excluded forest-minor. In this talk I will sketch a proof that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound exists if H has a cycle. Joint work with S. Fiorini and D. R. Wood. |
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25.04.2012 Gwenaël Joret |
Informatyka Teoretyczna Sorting under Partial Information (without the Ellipsoid Algorithm) |
| We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical. In this talk, we describe a simple and efficient algorithm for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithm differs in essential ways from theirs: Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed in a restricted class of graphs, permitting the use of a simpler algorithm. Furthermore, we compute or approximate the entropy at most twice, instead of doing it at each iteration. Joint work with J. Cardinal, S. Fiorini, R. M. Jungers, and J. I. Munro. |
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25.04.2012 Maciej Gawron |
Podstawy Informatyki COUNTING d-DIMENSIONAL POLYCUBES AND NONRECTANGULAR PLANAR POLYOMINOES by GADI ALEKSANDROWICZ and GILL BAREQUET |
A planar polyomino of size n is an edge-connected set of n squares on a rectangular two-dimensional lattice. Similarly, a d-dimensional polycube (for d 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d-1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier's algorithm for counting two-dimensional rectangular polyominoes, which counts all the above types of polyominoes. For example, our program computed the number of distinct three-dimensional polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value. |
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18.04.2012 Bartosz Szabucki |
Informatyka Teoretyczna Fast Minor Testing in Planar Graphs |
| Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C_u and C_v. A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time O(2^O(h)·n+n^2·log n) a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming. References:Isolde Adler, Frederic Dorn, Fedor V. Fomin, Ignasi Sau and Dimitrios M. Thilikos, Fast Minor Testing in Planar Graphs, Algorithmica, DOI 10.1007/s00453-011-9563-9 |
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18.04.2012 Patryk Zaryjewski |
Podstawy Informatyki Deciding if a Regular Language is Generated by a Splicing System by Lila Kari Steffen Kopecki |
| Splicing as a binary word/language operation is inspired by the DNA recombination under the action of restriction enzymes and ligases, and was first introduced by Tom Head in 1987. Shortly after, it was proven that the languages generated by (finite) splicing systems form a proper subclass of the class of regular languages. However, the question of whether or not one can decide if a given regular language is generated by a splicing system remained open. In this paper we give a positive answer to this question. We namely prove that if a language is generated by a splicing system, then it is also generated by a splicing system whose size is a function of the size of the syntactic monoid of the input language, and which can be effectively constructed. |
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12.04.2012 Jakub Przybyło (AGH) |
Algorytmiczne Aspekty Kombinatoryki Can colour-blind distinguish three-colour pallets? Yes they can! |
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11.04.2012 Robert Obryk |
Informatyka Teoretyczna Wait-free parallel summing snapshot |
| Atomic snapshot[1] is a well known parallel data structure that implements two operations: update which updates a thread's value and scan which returns an array of all threads' values. A wait-free implementation of this structure using O(1) time for update and O(n) time for scan in O(n) memory is known[2]. In this talk we will present an implementation of a similar structure, where scan returns not the whole array, but its sum (or the result of applying any other associative operation to its elements). The structure uses O(n) memory, update runs in O(log n) time and scan runs in O(1) time. We will also present an implementation of a structure, which has an atomic update-and-scan operation. Its memory complexity is O(n log^2 n), and time complexity of the operation is O(log^2 n). We will show how to implement that structure on machines with small word size without sacrificing wait-freeness nor complexity. References:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.3090&rep=rep1&type=pdf http://groups.csail.mit.edu/tds/papers/Shavit/RST.pdf |
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04.04.2012 Piotr Wójcik |
Informatyka Teoretyczna Algorithmic Meta-theorems for Restrictions of Treewidth |
| ``Possibly the most famous algorithmic meta-theorem is Courcelle's theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time's dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees. We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number.We prove stronger algorithmic meta-theorems for these more restricted classes of graphs.More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.'' References:Michael Lampis, Algorithmic Meta-theorems for Restrictions of Treewidth, Algorithmica, DOI 10.1007/s00453-011-9554-x |
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04.04.2012 Michał Feret |
Podstawy Informatyki Optimal Stopping and Applications 5 |
| Chapter 6. Maximizing the Rate of Return. |
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29.03.2012 Piotr Skowron (UW) |
Algorytmiczne Aspekty Kombinatoryki Selective complexity issues of elections |
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28.03.2012 Agnieszka Łupińska |
Podstawy Informatyki Optimal Stopping and Applications 4 |
| Chapter 5. Monotone Stopping Rule Problems. |
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22.03.2012 Paweł Petecki |
Algorytmiczne Aspekty Kombinatoryki Hamiltonian decompositions of k-uniform hypergraphs |
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21.03.2012 Mikołaj Bojańczyk, UW |
Informatyka Teoretyczna Automata Theory in XML |
| I will describe how finite automata are used in XML documents. The main idea is that an XML document is a finite tree, and therefore one can use tree automata to process XML documents. XML documents bring new questions that have not been previously studied by automata theory. Maybe the most interesting issue is that when modeling an XML document as a tree, the node labels come from an infinite alphabet. For instance, a node that stores a book in a bibliographic database comes with the book's ID. A typical query might check if the database contains two books with the same ID. |
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21.03.2012 Szymon Kałasz |
Podstawy Informatyki Optimal Stopping and Applications 3 |
| Chapter 3. THE EXISTENCE OF OPTIMAL STOPPING RULES. |
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15.03.2012 Arkadiusz Pawlik |
Algorytmiczne Aspekty Kombinatoryki Coloring intersection graphs of segments in the plane |
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15.03.2012 Paweł Petecki |
Algorytmiczne Aspekty Kombinatoryki Hamiltonian decompositions of k-uniform hypergraphs |
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14.03.2012 Bartłomiej Bosek,Grzegorz Matecki |
Informatyka Teoretyczna News on Semiantichains and Unichain Coverings |
| The famous Dilworth's Theorem states that for any partial order the size of a largest antichain is equal to the size of a smallest chain covering. In the late '70s C.Greene and D.Kleitman successfully generalized Dilworth's Theorem which moved forward the studies of partially ordered sets. Later, Saks proved equivalent statement that in the product CxQ of a chain C and an arbitrary poset Q the size of maximum antichain equals the size of minimum chain covering with chains of the form {c}xC' and Cx{q} (called unichains). We study Saks-West's conjecture which gives a nice generalization of the previous theorem: For any two posets P and Q the size of a maximum semiantichain and the size of minimum unichain covering in the product PxQ are equal. Here, semiantichain means a set for which each two points are not in a common unichain. B.Bosek, S.Felsner, K.Knauer and G.Matecki found conditions on P and Q that imply the conjecture in case of several new classes of posets. However, they also found an example showing that in general this min-max relation is false. This finally disproofs 30 year old conjecture of M.Saks and D.West. |
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14.03.2012 Paweł Wanat |
Podstawy Informatyki Optimal Stopping and Applications 2 |
| Chapter 4. Applications. Markov Models. |
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08.03.2012 Zbigniew Lonc (PW) |
Algorytmiczne Aspekty Kombinatoryki Constructions of asymptotically shortest k-radius sequences |
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07.03.2012 Kamil Kraszewski |
Informatyka Teoretyczna New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Applications |
| A pair of unit clauses is called conflicting if it is of the form (x), (¯x). A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28:411–421, 1981) showed that for each UCF CNF formula with m clauses we can simultaneously satisfy at least ϕ'm clauses, where ϕ'=(√5−1)/2. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula F with m clauses we can find, in polynomial time, a subformula F' with m' clauses such that we can simultaneously satisfy at least ϕ'm+(1−ϕ')m'+(2−3ϕ')n"/2 clauses (in F), where n"cis the number of variables in F which are not in F'.We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds m/2 and m(√5−1)/2. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31:335–354, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most 6k+3 variables and 10k clauses.We improve this to 4k variables and (2√5+4)k clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most (7+3√5)k variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above. References:Robert Crowston, Gregory Gutin, Mark Jones and Anders Yeo, A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Applications, Algorithmica DOI 10.1007/s00453-011-9550-1 |
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07.03.2012 Jonasz Pamuła |
Podstawy Informatyki Optimal Stopping and Applications 1 |
| First chapters of the book Optimal Stopping and Applications, by Thomas S. Ferguson. |
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01.03.2012 Monika Pilśniak (AGH) |
Algorytmiczne Aspekty Kombinatoryki Graph coloring for color blind persons |
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29.02.2012 Piotr Kołacz |
Informatyka Teoretyczna On-line approximate string matching with bounded errors |
| We introduce a new dimension to the widely studied on-line approximate string matching problem, by introducing an error threshold parameter ϵ so that the algorithm is allowed to miss occurrences with probability ϵ. This is articularly appropriate for this problem, as approximate searching is used to model many cases where exact answers are not mandatory. We show that the relaxed version of the problem allows us breaking the average-case optimal lower bound of the classical problem, achieving average case O(nlog_σ m/m) time with any ϵ=poly(k/m), where n is the text size,m the pattern length, k the number of differences for edit distance, and σ the alphabet size. Our experimental results show the practicality of this novel and promising research direction. Finally, we extend the proposed approach to the multiple approximate string matching setting, where the approximate occurrence of r patterns are simultaneously sought. Again, we can break the average-case optimal lower bound of the classical problem, achieving average case O(n log_σ(rm)/m) time with any ϵ=poly(k/m). References:Marcos Kiwi, Gonzalo Navarro and Claudio Telha, On-line approximate string matching with bounded errors, Theoretical Computer Science 412 (2011), 6359–6370 |
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29.02.2012 Jakub Kozik |
Podstawy Informatyki Property B - two coloring of uniform hypergraphs. |
| m(n) is defined to be the smallest number for which there exists an n-uniform hypergraph with m(n) edges which is not 2-colorable. Erdos and Lovasz conjectured that m(n) asymptotically behaves like n 2^n. I will present a simple proof of the best known lower bound sqrt(n/log(n)) 2^n, originally derived by Radhakrishnan and Srinivasan in 2000. |
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23.02.2012 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Multidimensional necklace splitting |
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18.01.2012 Jonasz Pamuła |
Informatyka Teoretyczna 1-Local 7/5-Competitive Algorithm for Multicoloring Hexagonal Graphs |
| In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by frequencies assigned to them, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where every vertex knows its position in the graph. We present a 1-local 7/5-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 17/12-competitive algorithm. References:Petra Šparl, Rafał Witkowski, Janez Žerovnik, 1-Local 7/5-Competitive Algorithm for Multicoloring Hexagonal Graphs, Algorithmica, DOI 10.1007/s00453-011-9562-x |
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18.01.2012 Michał Marczyk |
Podstawy Informatyki Unification through Projectivity by S. Ghilardi |
| We introduce an algebraic approach to E-unification, through the notions of finitely presented and projective object. As applications and examples, we determine the unification type of varieties generated by a single finite quasi-primal algebra, of distributive lattices and of some other equational classes of algebras corresponding to fragments of intuitionistic logic. |
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12.01.2012 Mateusz Nikodem |
Algorytmiczne Aspekty Kombinatoryki O wierzchołkowej stabilności grafów |
| Niech H będzie ustalonym grafem. Graf G nazywamy (H,k)-wierzchołkowo stabilnym jeśli G zawiera H nawet po usunięciu dowolnych k wierzchołków spośród V(G). Interesujące dla nas są grafy (H,k)-stabilne o możliwie małej liczbie krawędzi. |
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11.01.2012 Paweł Wanat |
Informatyka Teoretyczna Exact Algorithms for Edge Domination |
| An edge dominating set in a graph G=(V,E) is a subset of the edges D⊆E such that every edge in E is adjacent or equal to some edge in D. The problem of finding an edge dominating set of minimum cardinality is NP-hard. We present a faster exact exponential time algorithm for this problem. Our algorithm uses O(1.3226^n) time and polynomial space. The algorithm combines an enumeration approach of minimal vertex covers in the input graph with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwisely. In each of these refinement steps, the worst cases in the measure and conquer analysis of the current algorithm are reconsidered and a new branching strategy is proposed on one of these worst cases. In this way a series of algorithms appears, each one slightly faster than the previous one, ending in the O(1.3226^n) time algorithm. For each algorithm in the series, we also give a lower bound on its running time. We also show that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226^n) time and polynomial space using modifications of the algorithm for edge dominating set. In addition, we consider the matrix dominating set problem which we solve in O(1.3226^{n+m}) time and polynomial space for n×m matrices, and the parametrised minimum weight maximal matching problem for which we obtain an O∗(2.4179^k) time and space algorithm. References:Johan M.M. van Rooij, Hans L. Bodlaender, Exact Algorithms for Edge Domination, Algorithmica, DOI 10.1007/s00453-011-9546-x |
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11.01.2012 Michał Marczyk |
Podstawy Informatyki Unification through Projectivity by S. Ghilardi |
| We introduce an algebraic approach to E-unification, through the notions of finitely presented and projective object. As applications and examples, we determine the unification type of varieties generated by a single finite quasi-primal algebra, of distributive lattices and of some other equational classes of algebras corresponding to fragments of intuitionistic logic. |
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04.01.2012 Paweł Komosa |
Informatyka Teoretyczna An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion |
| The PATHWIDTH ONE VERTEX DELETION (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied FEEDBACK VERTEX SET (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et al. (WG, Lecture Notes in Computer Science, vol. 6410, pp. 196–207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7^k·n^{O(1)}. In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65^k·n^{O(1)}, thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp. 251–260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomassé (ACM Trans. Algorithms 6(2), 2010). References:Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk and Jakub Onufry Wojtaszczyk, An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion, Algorithmica DOI 10.1007/s00453-011-9578-2 |
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04.01.2012 Patryk Zaryjewski |
Podstawy Informatyki State complexity of power by Michael Domaratzki, Alexander Okhotin |
| The number of states in a deterministic finite automaton (DFA) recognizing the language L^k where L is regular language recognized by an n-state DFA, and k>=2 is a constant, is shown to be at most n2^((k-1)n) and at least (n-k)2^((k-1)(n-k)) in the worst case, for every n > k and for every alphabet of at least six letters. Thus, the state complexity of L^k is Θ(n2^((k-1)n)). In the case k=3 the corresponding state complexity function for L^3 is determined as (6n-3)/8 4^n - (n-1)2^n - n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be nk. This bound is shown to be tight over a two letter alphabet. |
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21.12.2011 14.12.2011,Dominik Dudzik |
Informatyka Teoretyczna Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs |
| The HAMILTONIAN CYCLE problem is the problem of deciding whether an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. Finding an exact algorithm that solves it in O*(α^n) time for some constant α<2 was a notorious open problem until very recently, when Björklund presented a randomized algorithm that uses O*(1.657^n) time and polynomial space. The LONGEST CYCLE problem, in which the task is to find a cycle of maximum length, is a natural generalization of the HAMILTONIAN CYCLE problem. For a claw-free graph G, finding a longest cycle is equivalent to finding a closed trail (i.e., a connected even subgraph, possibly consisting of a single vertex) that dominates the largest number of edges of some associated graph H. Using this translation we obtain two deterministic algorithms that solve the LONGEST CYCLE problem, and consequently the HAMILTONIAN CYCLE problem, for claw-free graphs: one algorithm that uses O*(1.6818^n) time and exponential space, and one algorithm that uses O*(1.8878^n) time and polynomial space. References:H.J. Broersma, F.V. Fomin, P. van 't Hof and D. Paulusma, Exact algorithms for finding longest cycles in claw-free graphs, Algorithmica, DOI 10.1007/s00453-011-9576-4 |
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21.12.2011 Dominik Dudzik |
Podstawy Informatyki Higher Order Matching and Games by Colin Stirling |
| Assume simply typed lambda calculus with base type 0 and the definitions of α-equivalence, β and η-reduction. A matching problem has the form v = u where v,u : A for some type A, and u is closed. The order of the problem is the maximum of the orders of the free variables x1,...,xn in v. A solution of a matching problem is a sequence of terms t1 ,..., tn such that v {t1/x1 ,..., tn/xn} =βη u. We provide a game-theoretic characterisation of higher-order matching. The idea is suggested by model checking games. We then show that some known decidable instances of matching can be uniformly proved decidable via the game-theoretic characterisation. |
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14.12.2011 Adam Zydroń |
Podstawy Informatyki Spanning forests on the Sierpinski gasket |
| We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd,b(n) with d = 2 and b = 3; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b. |
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07.12.2011 Wojciech Bukowicki |
Informatyka Teoretyczna Bipartite Matching in the Semi-streaming Model |
| We present the first deterministic 1+ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1/ε)^5) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(n polylog n) bits, where n=|V|. The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1/ε)^{1/ε}) passes. We show that even for bipartite graphs, McGregor's algorithm needs Ω(1/ε)^{Ω(1/ε)} passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=2/ε. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the i-th matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand. References:Sebastian Eggert, Lasse Kliemann, Peter Munstermann, Anand Srivastav, Bipartite Matching in the Semi-streaming Model, Algorithmica, DOI 10.1007/s00453-011-9556-8 |
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07.12.2011 Michał Handzlik |
Podstawy Informatyki SOME IMPROVEMENTS TO TURNER'S ALGORITHM FOR BRACKET ABSTRACTION by M. Bunder |
| A computer handles lambda terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm-the (fab) algorithm of Curry (see Curry and Feys [6])-is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases. A measure of the efficiency of an abstraction algorithm was first introduced by Kennaway as an upper bound of the length of the obtained combinatory term, as a function of the length of the original term and the number of variables to be abstracted. Mulder used these methods and experiments with many special cases, to compare the efficiency of the main algorithms listed above. The algorithm of Statman came out as the most efficient in the limiting case, but showed up as almost the worst in a number of reasonably simple special cases. Turner's algorithm was generally the best in these cases and was in fact Mulder's choice overall. In this paper, firstly we note that what Turner describes as "the improved algorithm of Curry", on which his own is based, is in fact not equivalent to any of Curry's algorithms. Turner's abstracts lack a basic property possessed by all of Curry's as well as many others. Secondly we give methods whereby Turner's algorithm (as well as others) can be more efficiently implemented, while providing simpler abstracts. |
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01.12.2011 Piotr Micek |
Algorytmiczne Aspekty Kombinatoryki Choice number versus its on-line counterpart |
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30.11.2011 Michał Feret |
Informatyka Teoretyczna Guard games on graphs |
| A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the guards and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is an induced subgraph of the given graph. We investigate the algorithmic aspects of the guarding problem, which is to find the minimum number of guards sufficient to patrol the area. We show that the guarding problem is PSPACE-hard and provide a set of approximation algorithms. All approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win. This variant of the game appears to be a 2-approximation for the guarding problem, and for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the one-step guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is W[1]-hard parameterized by the treewidth of the input graph. We also show that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including planar graphs and graphs of bounded genus, namely apex-minor-free graphs. We prove that the one-step guarding problem is FPT and possess a PTAS on apex-minor-free graphs. References:Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Guard games on graphs: Keep the intruder out! , Theoretical Computer Science 412 (2011), 6484–6497 |
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30.11.2011 Piotr Wójcik |
Podstawy Informatyki A note on propositional proof complexity of some Ramsey-type statements by Jan Krajicek |
| A Ramsey statement denoted n -> (k, 2, 2) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into valid DNF formula RAM(n, k) of size O(n^k) and with terms of size {k \choose 2}. Let r_k be the minimal n for which statement holds. We prove that RAM(r_k, k) requires expotential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll. |
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23.11.2011 Albert Łącki |
Informatyka Teoretyczna Almost Exact Matchings |
| In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m = m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. The first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable.We show how to count the number of exact perfect matchings in K_{3,3}-minor free graphs (these include all planar graphs as well as many others) in O(n^{3.19}) worst case time. Our algorithm can also count the number of perfect matchings in K_{3,3}-minor free graphs in O(n^{2.19}) time. References:Raphael Yuster, Almost Exact Matchings, Algorithmica, DOI 10.1007/s00453-011-9519-0 |
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16.11.2011 09.11.2011 02.11.2011 Lech Duraj Grzegorz Herman |
Informatyka Teoretyczna Garbage collection by reference counting the strongly connected components |
Traditional methods of automatic collection of unreachable objects (garbage) employ reference counting and/or graph traversal. The disadvantage of the former is its inherent inability to collect cyclic garbage structures (to deal with those, a supplemental, traversal-based method is often used). The latter suffers from having to traverse (at least periodically) all reachable objects. Heuristics based on the generational hypothesis lower this overhead, but only at the cost of failing to provide strong bounds on when garbage is going to be collected or how much total memory will be used. We propose a different approach, based on counting references between (approximations of) the strongly connected components of the object graph. Our method is real-time (has a constant time bound on any single operation), and concurrent (allows to interleave collection with program's activity). It provides an arbitrarily strong bound on memory usage. It makes use of the generational hypothesis, avoiding the traversal of objects permanently reachable by "old enough" edges. It uses constant memory per managed object plus constant global memory, and employs no data structures more complex than a stack, which gives hope that it can be performed in a lock-free manner. |
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16.11.2011 Michał Masłowski |
Podstawy Informatyki Coarse-graining of cellular automata, emergence, and the predictability by Navot Israeli, Nigel Goldenfeld |
| Using nearest neighbor, one-dimensional Cellular Automata (CA), we show how to construct local coarse-grained escriptions of CA with different complexity classification. Large-scale behavior can be emulated by them without small-scale details. We show that coarse-grained CA can be decidable even for universal original systems and coarse-graining transformations make a complexity hierarchy of CA rules. Finally we argue that the large scale dynamics of CA can have very small Kolmogorov complexity of the update rules, and moreover exhibits a novel scaling law. This makes finding coarse-grained descriptions of CA easier at coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics. |
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10.11.2011 Bartosz Walczak |
Algorytmiczne Aspekty Kombinatoryki Outerplanar graph drawings with few slopes |
| How many different edge slopes are necessary and sufficient to draw any outerplanar graph of degree Delta in the plane in the outerplanar way, that is, so that edges are non-crossing straight-line segments and all vertices lie on the outer face? We show that this number for Delta>3 is precisely Delta-1. This is joint work with Kolja Knauer and Piotr Micek. |
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09.11.2011 Marek Wróbel |
Podstawy Informatyki Complexity of Type Inference by Jerzy Tyszkiewicz |
| The type inference problem may be stated as follows: given a term of untyped lambda calculus, decide whether it may be typed to a term of a first-order-typed lambda calculus. If it is possible, then find all possible typings for it (or at least one possible typing). We show that the type inference problem is PTIME-complete. |
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03.11.2011 Arkadiusz Pawlik |
Algorytmiczne Aspekty Kombinatoryki On the Albertson Crossing Conjecture |
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02.11.2011 Przemysław Jedynak |
Podstawy Informatyki How common can be universality in cellular automata? by Guillaume Theyssier |
| We address the problem of the density of intrinsically universal cellular automata among cellular automata or a subclass of cellular automata. We show that captive cellular automata are almost all intrinsically universal. We show however that intrinsic universality is undecidable for captive cellular automata. Finally, we show that almost all cellular automata have no non-trivial sub-automaton. |
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26.10.2011 19.10.2011,Michał Staromiejski |
Informatyka Teoretyczna When are finite rings indecomposable? |
| The main goal of the talk is to present a simple characterization of local (a.k.a. indecomposable) finite algebras over (finite) fields. The characterization leads to polynomial-time algorithm for testing locality of such algebras and, in turn, to polynomial-time locality test for arbitrary finite rings. Generalization for algebras over arbitrary fields and related open questions will be discussed. |
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26.10.2011 Maciej Bendkowski |
Podstawy Informatyki On Post correspondence problem for letter monotonic languages by Vesa Halava, Jarkko Kari, Yuri Matiyasevich |
| We prove that for given morphisms g, h: { a_1, ..., a_n } \to B^{*}, it is decidable whether or not there exists a word w in the regular language a_{1}^{*}, ..., a_{n}^{*} such that g(w) = h(w). In other words, we prove that the Post Correspondence Problem is decidable if the solutions are restricted to be from this special language. This yields a nice example of an undecidable problem in integral matrices which cannot be directly proved undecidable using the traditional reduction from the Post Correspondence Problem. |
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19.10.2011 12.10.2011,Robery Obryk |
Podstawy Informatyki Dowodzenie w językach z typami zależnymi |
| Celem systemów typów w językach programowania jest ułatwianie wnioskowania o poprawności kodu. Już języki używające systemu typów Hindleya-Milnera pozwalają wnioskować w ciekawy sposób o zachowaniu funkcji na podstawie ich typu. Języki z typami zależnymi pozwalają wyrażać za pomocą typu funkcji praktycznie arbitralne warunki na jej zachowanie. Na tym seminarium poznamy formalizm typów zależnych, sposób przeprowadzania izomorfizmu Curryego-Howarda w nim i przyjrzymy się jak typy zależne pozwalają dowodzić własności programu i pomagają w dowodzeniu własności stopu w języku Agda. |
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12.10.2011 05.10.2011,Leszek Horwath |
Informatyka Teoretyczna Asymptotically Optimal Randomized Rumor Spreading |
| New protocol solving the fundamental problem of disseminating a piece of information to all members of a group of n players. |
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05.10.2011 Robert Obryk |
Podstawy Informatyki Dowodzenie w językach z typami zależnymi |
| Celem systemów typów w językach programowania jest ułatwianie wnioskowania o poprawności kodu. Już języki używające systemu typów Hindleya-Milnera pozwalają wnioskować w ciekawy sposób o zachowaniu funkcji na podstawie ich typu. Języki z typami zależnymi pozwalają wyrażać za pomocą typu funkcji praktycznie arbitralne warunki na jej zachowanie. Na tym seminarium poznamy formalizm typów zależnych, sposób przeprowadzania izomorfizmu Curryego-Howarda w nim i przyjrzymy się jak typy zależne pozwalają dowodzić własności programu i pomagają w dowodzeniu własności stopu w języku Agda. Tematy do nastęnych seminariów: "A Predicative Analysis of Structural Recursion" Abel, Altenkirch "The Size-Change Principle for Program Termination" Lee, Jones, Ben-Amran |
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09.06.2011 Jakub Przybyło (AGH) |
Algorytmiczne Aspekty Kombinatoryki Proper edge colourings with different color paletts on adjacent vertices - algorithms based on Combinatorial Nullstellensatz |
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02.06.2011 Paweł Rzążewski (Politechnika Warszawska) |
Algorytmiczne Aspekty Kombinatoryki Exact algorithms for L(2,1)-coloring problem |
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01.06.2011 Kamil Kraszewski |
Informatyka Teoretyczna A New Upper Bound for Max-2-SAT: A Graph-Theoretic Approach |
| In MaxSat, we ask for an assignment which satisfies the maximum number of clauses for a boolean formula in CNF. We present an algorithm yielding a run time upper bound of O*(2^(K/6.2158)) for Max-2-SAT (each clause contains at most 2 literals), where K is the number of clauses. The run time has been achieved by using heuristic priorities on the choice of the variable on which we branch. The implementation of these heuristic priorities is rather simple, though they have a significant effect on the run time. Also the analysis uses a non-standard measure time. References:D. Raible, H. Fernau. A new upper bound for max-2-sat: A graph-theoretic approach. MFCS, LNCS 5162, 551–562, 2008 |
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01.06.2011 Małgorzata Kruszelnicka |
Podstawy Informatyki Bisymulation on finite Kripke models |
| The notion of bisimulation has been introduced to test whether two processes behave the same. Originally discovered in Computer Science, bisimulation nowadays is employed in many fields. Today it is used in a number of areas of Computer Science such as functional languages, data types, databases, program analysis, to name but a few. Growing interests in this notion led to the discovery of bisimulation in Modal Logic and Set Theory. Finally, the notion was introduced into first-order logic, and found a straightforward game-theoretical interpretation. We present the notion of bisimulation for intuitionistic logic. Our discussion focuses on two cases: the propositional and the first-order case. In both cases we present the theorem which states that bisimulation implies logical equivalence, and consider possible variants of the inverse implication. Further, we present our contribution to the research of the inverse theorem. References: Patterson, A.: \emph{Bisimulation and Propositional Intuitionistic Logic}, Proceedings of the 8th International Conference on Concurrency Theory, Springer-Verlag, 1997 Po{\l}acik, T.: \emph{Back and Forth Between First-Order Kripke Models}, Logic Journal of the IGPL 16 (4), 335--355, 2008 Visser, A.: \emph{Bisimulations, Model Descriptions and Propositional Quantifiers}, Logic Group Preprint Series 161, 1996 |
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26.05.2011 Paweł Petecki (AGH) |
Algorytmiczne Aspekty Kombinatoryki Grasshopper jumping in both directions |
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25.05.2011 Agnieszka Łupińska |
Podstawy Informatyki A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points, by Noson S. Yanofsky |
| kontynuacja |
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18.05.2011 Marek Wróbel |
Informatyka Teoretyczna Design and analysis of online batching systems |
| References:Regant Y.s. Hung, Hing-Fung Ting, Design and analysis of online batching systems, Algorithmica (2010), 57: 217--231 |
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18.05.2011 Agnieszka Łupińska |
Podstawy Informatyki A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points, by Noson S. Yanofsky |
| Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory. |
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11.05.2011 Robert Obryk |
Podstawy Informatyki Algorithmic Information Theory 2, by Gregory. J. Chaitin |
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05.05.2011 A. Nilli |
Algorytmiczne Aspekty Kombinatoryki On the chromatic number of random Cayley graphs |
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04.05.2011 Michał KukiełaUMK |
Informatyka Teoretyczna Some combinatorial approaches to homotopy |
| Different notions of "homotopy" equivalences of partially ordered sets may be defined in terms of various one-point reductions and expansions. These have been a recent object of study of J.A. Barmak and G.E. Minian. Their research was inspired by results from the 60's of R.E. Stong, who classified, using elementary "deformations", the homotopy types of finite topological spaces. Finite spaces satisfying the T0 separability axiom may be easily identified with partially ordered sets, and the deformations of Stong turn out to be dismantlings by irreducible points. Some, natural from a topologist's point of view, generalizations of irreducible points give interesting definitions of "homotopy". I will present relations between these notions and their connections to topics such as poset fixed point theory, evasiveness and homotopy theory of polyhedra. |
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04.05.2011 Robert Obryk |
Podstawy Informatyki Algorithmic Information Theory , by Gregory. J. Chaitin |
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27.04.2011 Piotr Wójcik |
Informatyka Teoretyczna An Approximation Algorithm for Binary Searching in Trees |
| We consider the problem of computing efficent strategies for searching in trees. As a generalization of the classical binary search for ordered lists, suppose one wishes to find a (unknown) specific node of tree by asking queries to its arcs, where each query indicates the endpoint closer to the desired node. Given the likelihood of each node being the one searched, the objective is to compute a search strategy that minimizes the expected number of queries. Practical applications of this problem include file system synchronization and software testing. Here we present a linear time algorithm which is the first constant factor approximation for this problem. This represents a significant improvement over previous O(log n) approximation. References:Eduardo Laber and Marco Molinaro, An Approximation Algorithm for Binary Searching in Trees, Algorithmica, 59(2010), 601-620 |
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27.04.2011 20.04.2011,Marek Wróbel, Adam Zydroń |
Podstawy Informatyki Mathematics for the Analysis of Algorithms - Asymptotic Analysis |
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20.04.2011 Piotr Szafruga |
Informatyka Teoretyczna Greedy Remote-Clique Algorithm |
| References:B.Birnbaum and K.J.Goldman, An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs |
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13.04.2011 Grzegorz Guśpiel |
Informatyka Teoretyczna A Constant Space, Subquadratic Algorithm for Inverse Permutation |
| We assume the permutation is given by an array in which the i-th element denotes the value at i. Finding its inverse can be achieved in linear time with a simple cycle-based algorithm. Limiting the numbers that can be stored in our array to the range of the permutation still allows a simple O(n^2) solution. A better O(n^{3/2}) algorithm will be presented. |
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13.04.2011 06.04.2011,Maciej Bendkowski |
Podstawy Informatyki Mathematics for the Analysis of Algorithms - Operator Methods |
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30.03.2011 Marek GrabowskiUW |
Informatyka Teoretyczna Computing steady state of Markov Chain by combinatorial aggregation |
| Probabilistic model checking is receiving quite a lot of attention around the world recently (i.e. DARPA is funding PRISMATIC project). Unlike 'normal' model checking which was research for nearly 30 years, probabilistic model checking is still a young discipline. Most common framework for modeling probabilistic processes are Markov Chains, both discrete and continuos time and Markov Decision Processes. One of interesting questions one can ask about DTMCs and CTMCs is 'to what distribution given chain converges' (what's the steady state of it). Theory of Markov Chains has over 100 years now and analytic solutions of all interesting questions are well known. Problem with such solutions is that they're usually untraceable for real-life models, because of their size. This is the reason why iterative methods are most commonly used. Unfortunately they also fail for some examples. I'll show an algorithm which was proposed by Pokarowski in his PhD thesis and implemented by me just recently, which for some class of models gives huge speedup in return for some precision. I'll describe theory behind this algorithm, show how it works in general and some test results. I'll also tell what modifications are on the way and what we hope to achieve in the end. |
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23.03.2011 Maciej Wawro |
Informatyka Teoretyczna Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs |
| Motivated by applications in batch scheduling of jobs in manufacturing systems and distributed computing, we study two related problems. Given is a set of jobs {J1, . . . , Jn}, where Jj has the processing time pj , and an undirected intersection graph G = ({1, 2, . . . , n}, E), with an edge (i, j) whenever the pair of jobs Ji and Jj cannot be processed in the same batch. We are to schedule the jobs in batches, where each batch completes its processing when the last job in the batch completes execution. The goal is to minimize the sum of job completion times. Our two problems differ in the definition of completion time of a job within a given batch. In the first variant, a job completes its execution when its batch is completed, whereas in the second variant, a job completes execution when its own processing is completed.
For the first variant, we show that an adaptation of the greedy set cover algorithm gives a 4-approximation for perfect graphs. For the second variant, we give new or improved approximations for a number of different classes of graphs. The algorithms are of widely different genres (LP, greedy, subgraph covering), yet they curiously share a common feature in their use of randomized geometric partitioning. References:Leah Epstein, Magnus M. Halldórsson, Asaf Levin, Hadas Shachnai, Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs , Algorithmica 55(2009) 643-665 |
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23.03.2011 Anna Bień (UŚ) |
Podstawy Informatyki Klasy grafów singularnych |
| We consider simple graphs and their adjacency matrices. In [1] Rara gives graphs with singular adjacency matrix are called singular. In [1] Rara presented tools, which are useful in computing determinant of adjacency matrix of some simple graphs. Rara's methods allow to replace complicated algebraic calculations with operations performed on graphs. In some cases removing sets of edges or vertices does not change or changes the determinant of a graph in a specific way. We continue this subject matter and present general methods of reducing graphs. The most general is the method of identifying $P_3$ paths. Consequences of this theorem are the method of contracting $P_5$ path and a method which can be applied to graphs circumscribed on cycles. We apply these methods in computing determinant of adjacency matrix of certain classes of graphs. In particular we present a recursive formula for planar grids $P_n \times P_{n+1}$ $$det A(P_n \times P_{n+1})= (-1)^{(n+1)/2}$$ which is a main step in solution of the singularity problem for all planar grids.
[1] H.M. Rara, Reduction procedures for calculating the determinant of the adjacency matrix of some graphs and the singularity of square planar grids, Discrete Mathematics 151, 213-219, Elsevier, 1996. |
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17.03.2011 Michał Lasoń |
Algorytmiczne Aspekty Kombinatoryki Mr. Paint and Mrs. Correct are doing it on matroids |
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16.03.2011 09.03.2011,Michał Masłowski |
Podstawy Informatyki Mathematics for the Analysis of Algorithms - Recurrence Relations |
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10.03.2011 Przemysław Mazur |
Algorytmiczne Aspekty Kombinatoryki Multiple recurrence and arithmetic progressions |
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09.03.2011 Andrzej Grzesik |
Informatyka Teoretyczna Maximum number of pentagons in triangle free graphs |
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03.03.2011 Michał Farnik |
Algorytmiczne Aspekty Kombinatoryki Cuts, Graphons, and Szemeredi Regularity Lemma |
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02.03.2011 Tomasz Bińczycki |
Podstawy Informatyki Mathematics for the Analysis of Algorithms - Binomial Identities |
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26.01.2011 19.01.2011,Michał Feret |
Informatyka Teoretyczna Fast 3-coloring triangle free planar graphs |
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19.01.2011 Przemek Jedynak |
Podstawy Informatyki Synthetic Differential Geometry - Chicago's pizza seminar notes. |
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13.01.2011 Marcin Witkowski |
Algorytmiczne Aspekty Kombinatoryki Nonrepetitive sequences up to (mod k) |
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12.01.2011 Tomasz Krakowiak |
Podstawy Informatyki Complexity of Type Inference, paper by Jurek Tyszkiewicz |
| The main result is the proof of PTIME-completeness of the type reconstruction problem for simply typed lambda calculus. |
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05.01.2011 Adam Zydroń |
Informatyka Teoretyczna Improved Parameterized Set Splitting Algorithms: A Probabilistic Approach |
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05.01.2011 Patryk Zaryjewski |
Podstawy Informatyki Interaction properties of relational periods, paper by Vesa Halava and Tero Harju and Tomi K¨arkiy |
| We consider relational periods where the relation is a compatibility relation on words induced by a relation on letters. We introduce three types of periods, namely global, external and local relational periods, and we compare their properties by proving variants of the theorem of Fine and Wilf for these periods. |
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16.12.2010 Arkadiusz Pawlik |
Algorytmiczne Aspekty Kombinatoryki Coloring geometric intersection graphs |
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15.12.2010 Przemysław Derengowski |
Informatyka Teoretyczna A best online algorithm for scheduling on two parallel batch machines. |
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15.12.2010 Jacek Bąkowski |
Podstawy Informatyki On systems of word equations ..., paper by Elena Czeizler, and Wojciech Plandowski |
| In this paper, we investigate the open question, formulated in 1983 by Culik II and Karhumäki, asking whether there exist independent systems of three word equations over three unknowns admitting non-periodic solutions. In particular, we answer negatively the above mentioned question for systems in which one of the unknowns occurs at most six times. That is, we show that such systems admit only periodic solutions or they are not independent. |
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09.12.2010 Michał Zmarz |
Algorytmiczne Aspekty Kombinatoryki Graph layouts and nonrepetitive colorings |
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08.12.2010 Michał Handzlik |
Informatyka Teoretyczna A fully dynamic graph algorithm for recognizing proper interval graphs |
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08.12.2010 Hannes Diener (Univ. of Siegen, Germany) |
Podstawy Informatyki Variations on a theme by Ishihara |
| This will be a talk in two halves. The first will consist of a gentle introduction to the area of constructive analysis, especially focussing on continuity and completeness. In constructive mathematics one is interested in objects that one cannot only rule out the non-existence of, but those that one can (at least in theory) actually construct. This part of the talk should be accessible to anyone with a knowledge of basic analysis - no knowledge about constructivism is required. In the second half we will present results by Hajime Ishihara of 1991, which became known as ``Ishihara's tricks''. In these results decisions that, on first and maybe even second glance, seem algorithmically impossible are made. We will present new results, which extend Ishihara's ideas to a more general setting. Lastly, we will show how all of this can be used to give an axiomatic, concise, and clear proof of the well known phenomenon that in many constructive settings every real-valued function on the unit interval is continuos (``computability implies continuity''). |
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02.12.2010 Leszek Horwath |
Algorytmiczne Aspekty Kombinatoryki On-line conflict-free coloring of intervals |
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01.12.2010 Marcin WitkowskiUAM |
Informatyka Teoretyczna Load balancing games |
| Load balancing games are the following kind of problems. Assume we are given M machines and N jobs. Each job i is associated with a vector p=(p_1,..,p_m), where p_j is the processing time of this job on machine j. Players correspond to jobs. The strategy set of each player is the set of machines. Given a strategy for each player, the total load on each machine is the sum of processing times of the jobs that chose that machine. The aim of each player is to minimize the total load on its chosen machine. We, as an external observer, are interested in minimizing the total load on the most-loaded machine. We call a Nash Equilibriun (NE), a strategy profile (vector consist of choices of each player) that is resilient to unilateral deviations, which means that no player has anything to gain by changing only his or her own strategy unilaterally. A downside of NE is that it is not necessarily stable against deviations by coalitions of players. A pure Nash equilibrium which is resilient to deviations by coalitions is called a strong equilibrium (SE). Using a framework introduced by Feldman and Tamir [1] I estimate how close a NE is to SE in certain load balancing games. References:[1] M. Feldman and T. Tamir. ,,Approximate strong equilibrium in job scheduling games". In SAGT, pages 58-69, 2008. |
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01.12.2010 Paweł Błasiak (IFJ -Kraków) |
Podstawy Informatyki Combinatorial Models of Creation-Annihilation |
| Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. We surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable graphs, or ''diagrams'', that are composed of elementary ''gates''. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, etc. Reference: P. Blasiak and Ph. Flajolet "Combinatorial Models of Creation-Annihilation" arXiv:1010.0354 [math.CO] |
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24.11.2010 Jakub Kozik |
Informatyka Teoretyczna Secretary problem on partial orders. |
| n secretaries apply for a single secretary position. All the candidates are partially ordered according to their competences. They are interviewed one by one in a random order. After each interview we know only the induced partial order on the secretaries interviewed so far. The decision whether to accept or reject a candidate must be made just after the interview. The objective is to choose one of the maximal candidates. We are going to present some recent result in the search for optimal strategy, with special emphasis on the result by R. Freji and J. Wästlund whose strategy achieves probability of success 1/e on every partial order. References:Freij, Ragnar; Wästlund, Johan; Partially ordered secretaries; Electronic Communications in Probability; Vol. 15 (2010) paper 45, pages 504-507. |
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24.11.2010 Michał Handzlik |
Podstawy Informatyki Pseudotopological spaces and the Stone-Cech compactification. |
| Our slogan is "topology is about convergence". The fact that so many aspects of topology can be captured by convergence naturally makes us wonder whether convergence could be taken to be more fundamental than open sets. It leads to the definition od pseudotopological space, where we focus on convergence properties only, without mentioning open sets. We present how Stone-Cech compactification translates into pseudotopological spaces. |
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18.11.2010 Marek Adrian |
Algorytmiczne Aspekty Kombinatoryki Covering systems of congruences; an application of the number-theoretic local lemma |
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17.11.2010 Jonasz Pamuła |
Podstawy Informatyki Algorithms for learning regular expressions from positive data, paper by Henning Fernau |
| We describe algorithms that directly infer very simple forms of 1-unambiguous regular expressions from positive data. Thus, we characterize the regular language classes that can be learned this way, both in terms of regular expressions and in terms of (not necessarily minimal) deterministic finite automata. |
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10.11.2010 Bartosz Walczak |
Informatyka Teoretyczna An extremal problem for crossing vectors |
| Two vectors u,v ∈ Zw are called crossing if there are two coordinates i,j such that ui−vi ≥ 1 and vj−uj ≥ 1. They are called k-crossing if there are two coordinates i,j such that ui−vi ≥ k and vj−uj ≥ k. We consider the following question: what is the maximum size of a family of pairwise crossing but not k-crossing vectors in Zw? Several (totally different) constructions of such families of size kw−1 are known. The conjecture is that kw−1 is optimal. The problem has been posed by Felsner and Krawczyk in February 2010. Since then several groups have been trying hard to solve it, with only little success. The conjecture is trivial for w=1,2. I will present the proof for w=3. It is not clear whether it brings us closer to the proof of the general conjecture, but it seems promising. For w≥4 the question is open. Solving the problem for general w might give us some new insights into posets and Sperner theory. |
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10.11.2010 Ola Piktus |
Podstawy Informatyki Topology on words, paper by Cristian S. Calude, Helmut Jürgensen, Ludwig Staiger |
| We investigate properties of topologies on sets of finite and infinite words over a finite alphabet. The guiding example is the topology generated by the prefix relation on the set of finite words, considered as a partial order. This partial order extends naturally to the set of infinite words; hence it generates a topology on the union of the sets of finite and infinite words. We consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural. We provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion. |
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03.11.2010 27.10.2010,Grzegorz Matecki |
Informatyka Teoretyczna First-Fit coloring of co-comparability graphs |
| One of the simplest heuristics to obtain a proper coloring of a graph is First-Fit algorithm. First-Fit visits each vertex of a graph in the already given order and assigns to it the smallest possible number (color) such that two vertices connected by an edge are not monochromatic. The class of 2-colorable co-comparability graphs is known to be infinitely colorable by First-Fit. We proved that H-free co-comparability graphs with a fixed chromatic number are finitely colorable by First-Fit if and only if H is a 2-colorable co-comparability graph. It provides the full characterization of First-Fit on co-comparability graphs in terms of forbidden structures. This is a joint work with Bartłomiej Bosek and Tomasz Krawczyk. |
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03.11.2010 27.10.2010,Kasia Grygiel |
Podstawy Informatyki Asymptotically almost all lambda terms are strongly normalizing |
| We present quantitative analysis of various (syntactic and behavioral) properties of random lambda terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the lambda calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator. |
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28.10.2010 Karol Kosiński |
Algorytmiczne Aspekty Kombinatoryki On some edit distance problems |
| String edit distance is a minimum total cost of edit operations (inserting, deleting and changing letters) needed to receive one string from another. Edit distance problem can be also extendeded in many ways to rooted labeled trees. We will consider constrained variants of tree edit distance problem and related tree inclusion problem in order to show efficient solutions to some classes of mentioned trees. The seminar is based on my master's thesis. |
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20.10.2010 13.10.2010,Marek Adrian |
Informatyka Teoretyczna Contributions of Endre Szemerédi in theoretical computer science |
| This talk is based on one given by Avi Wigderson presented on a conference honoring of the 70th birthday of Endre Szemerédi. Out of several results we will look closely into three proofs Szemerédi participated in. At first we will check the Dictionary Problem. We want to store a set U = {u1, … , un} subset 2k (n<<2k) using O(n) time & space. The question is how to minimize the number of queries to determine if x is in U. Next we shall look at Sorting Network where one using O(nlogn) comparators has been explicitly given. Finally we shall compare deterministic and non deterministic algorithms in linear time and their impact on k-paged graphs. |
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20.10.2010 Piotr Faliszewski (AGH) |
Podstawy Informatyki A 2-Approximation Algorithm for a Candidate Promotion Problem |
| We are given an election E=(C,V), where C is a set of alternatives, and V is a collection of voters, and a preferred alternative p. Each voter is represented by a linear order over C. As part of a political campaign, we have the ability (at some cost) to shift p forward in some of the votes. In the shift-bribery problem we ask for the minimal cost of shifts that ensure our candidate's victory. We show that this problem is NP-complete (for Borda winner determination rule; an example of so-called scoring rules) and give a 2-approximation algortihm that works for all scoring rules, even if the votes are weighted. |
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07.10.2010 Maciej Ulas |
Algorytmiczne Aspekty Kombinatoryki Arithmetic properies of Stern polynomials |
| We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. We study also the sequence $e(n)=\op{deg}_{t}B_{n}(t)$ and give many of its properties. This is joint work with Oliwia Ulas. |
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06.10.2010 Torsten UeckerdtTechnical University Berlin |
Informatyka Teoretyczna Intersection graphs of gridpaths |
| We are considering so called EPG representations of simple graphs, that is every vertex is modeled by a path in the plane square grid, such that the paths of two vertices have a grid-edge in common iff the two vertices are adjacent. The bend-number b(G) of a graph G is the minimal number k, such that G has an EPG representation with each path having no more than k bends. The bend-number is related to a graph's interval-number and track-number. For certain graph classes (planar graphs, complete bipartite graphs, graphs with maximum degree D, ...) we are now interested in the maximum interval-, track-, and bend-number among all graphs in the class. We settle some answers but still are left with many open questions. |
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10.06.2010 Andrzej Lembas |
Algorytmiczne Aspekty Kombinatoryki Elections can be manipulated often |
| Every non-trivial voting method between at least 3 alternatives can be strategically manipulated. Definitions of Social choice functions, manipulation, manipulation power. Main theorem: There exists a constant C>0 such that for every e>0, if f is a neutral social choice function among 3 alternatives for n voters, then the sum of probabilities manipulation power is >= Ce^2. Example. |
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09.06.2010 Tomasz Krawczyk |
Informatyka Teoretyczna On-line dimension for posets excluding two long chains |
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02.06.2010 Maciej Wawro |
Informatyka Teoretyczna Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs |
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27.05.2010 Tibor SzaboFreie Universitat Berlin |
Algorytmiczne Aspekty Kombinatoryki On minimal Ramsey graphs |
| A graph G is called H-Ramsey if any two-coloring of the edges of G contains a monochromatic copy of H. An H-Ramsey graph is called H-minimal if no proper subgraph of it is H-Ramsey. Minimal Ramsey graphs were the subject of intensive research in the past four decades, going back to an innocent looking question of Erdos and Hajnal from the 60's concerning the existence of K_6-free K_3-Ramsey graphs. After an overview of the topic we concentrate on the minimum degree of H-minimal graphs, a problem initiated by Burr, Erdos, and Lovasz. We determine the smallest possible minimum degree of H-minimal graphs for numerous bipartite graphs H, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number. This represents joint work with Philipp Zumstein and Stefanie Zurcher. |
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26.05.2010 Mateusz Drewienkowski |
Informatyka Teoretyczna Priority algorithms for graph optimization problems |
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13.05.2010 Zofia Miechowicz University of Grunberg |
Algorytmiczne Aspekty Kombinatoryki Game chromatic number of Cartesian product graphs |
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12.05.2010 Przemysław Gordinowicz TU Łódź |
Informatyka Teoretyczna On graphs isomorphic to their neighbour and non-neighbour sets |
The talk describes a construction of a universal countable graph, different from the Rado graph, such that for any of its vertices both the neighbourhood and the non-neighbourhood induce subgraphs isomorphic to the whole graph. This solves an open problem posed by A. Bonato at 18th BCC |
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05.05.2010 Szymon Borak |
Informatyka Teoretyczna Optimal algorithms for the path/tree-shaped facility location problems in trees |
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29.04.2010 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki List Edge Colourings |
| I will present the classical result of Galvin settling the famous list colouring conjecture for bipartite graphs. Some related problems and questions will be posed. Based on a paper of Borodin, Kostochka and Woodall. References:O.V.Borodin, A.V.Kostochka, D.R.Woodall, List Edge and List Total Colourings of Multigraphs, Journal of Combinatorial Theory, Series B 71, pp. 184-204, 1997 |
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28.04.2010 Kasper Kopeć |
Informatyka Teoretyczna Finding paths between graph colorings: PSPACE-completeness |
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15.04.2010 canceled |
Algorytmiczne Aspekty Kombinatoryki |
| In view of an alternative event: http://www.ii.uj.edu.pl/index.php?mact=News,cntnt01,detail,0&cntnt01articleid=37&cntnt01returnid=92&hl=pol |
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14.04.2010 Rafał Pajdzik |
Podstawy Informatyki The Normalization Theorem, Chapter 4 of Girard's book. |
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08.04.2010 Witold SzczechlaWarsaw University |
Algorytmiczne Aspekty Kombinatoryki A solution for the three-colour hat guessing problem for cycles |
| We consider the hat guessing problem for cycles. We prove that Bears win the 3-colour game on C_N if and only if N is divisible by 3, or N=4. |
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07.04.2010 31.03.2010,Kamil Kraszewski |
Informatyka Teoretyczna Preemptive online scheduling: optimal algorithms for all speeds |
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07.04.2010 Leszek Horwath (Cedric) |
Podstawy Informatyki Curry Howard Isomorphism. Chapter 3 of Girard's book. |
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25.03.2010 Mikołaj Pudo |
Algorytmiczne Aspekty Kombinatoryki Geometry of solution space of random SAT |
| In this talk, I will discuss the structure of satisfying assignments of a random k-SAT formula. We will be interested in the evolution of geometry of this space as constraints (clauses) are added. In particular I will show that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. |
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24.03.2010 17.03.2010,Piotr Micek |
Informatyka Teoretyczna Algorithmic version of the Lovász Local Lemma |
| We will study the new approach of Robin Moser to give an algorithm for the Lovász Local Lemma. Most likely, we will stick to symmetric case. References:R. A. Moser, G. Tardos - A constructive proof of the general Lovász Local Lemma |
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18.03.2010 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki The Lovasz Local Lemma - satisfaction guaranteed |
| I will present some recent spectacular applications of the local lemma for various problems in distinct areas of Mathematics and Computer Science. New directions for further research will be discussed. |
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11.03.2010 Piotr Szafruga |
Algorytmiczne Aspekty Kombinatoryki Algebraic proof of Brooks' theorem |
| I will present proof of Brooks' theorem for list coloring using the algebraic method of Alon and Tarsi. This also shows that the Brooks' theorem remains valid in more general game coloring setting. Based on paper of Hladky, Kral and Schauz. |
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04.03.2010 Michał Lasoń |
Algorytmiczne Aspekty Kombinatoryki Algebraic methods in discrete analogs of the Kakeya problem |
| I will prove the "joints" conjecture, which says that given N lines in space there are at most O(N^(3/2)) joints, that is points where at least three not coplanar lines intersect. Based on paper of Guth and Katz. |
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25.02.2010 Wesley PegdenRutgers University |
Algorytmiczne Aspekty Kombinatoryki Games where the Local Lemma works |
| In this talk, I will discuss probabilistic proofs for the existence of winning strategies in sequence games where the goal is nonrepetitiveness. The technique involves a `one-sided' generalization of the Local Lemma, which allows us to ignore the dependencies on `future' events which would normally prevent this kind of proof from working. I will also discuss the extension of these results to graphs. Although many proofs about games are motivated by a probabilistic intuition, these results appear to represent the first successful applications of a Local Lemma to combinatorial games. If there is time, I will discuss an interesting (and frustrating!) game where this kind of approach has not yet succeeded. |
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28.01.2010 ZAJĘCIA W NOWYM SEMESTRZE |
Podstawy Informatyki PROOFS AND TYPES |
| Link do książki Girarda pt. Proofs and Types. List of chapter titles: 1. Sense, Denotation and Semantics 2. Natural Deduction 3. The Curry-Howard Isomorphism 4. The Normalisation Theorem 5. Sequent Calculus 6. Strong Normalisation Theorem 7. Gödel's system T 8. Coherence Spaces 9. Denotational Semantics of T 10. Sums in Natural Deduction 11. System F 12. Coherence Semantics of the Sum 13. Cut Elimination (Hauptsatz) 14. Strong Normalisation for F 15. Representation Theorem Appendices: A. Semantics of System F - by Paul Taylor B. What is Linear Logic? - by Yves Lafont |
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27.01.2010 Marek Wróbel |
Podstawy Informatyki Program Analysis |
| Ostatnia część książki Changa and Lee o dowodzeniu twierdzeń. |
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20.01.2010 Michał Handzlik |
Podstawy Informatyki Procedury dowodowe oparte na twierdzeniu Herbranda |
| Rozdział 9. książki Changa i Lee o automatycznym dowodzeniu twierdzeń. |
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13.01.2010 06.01.2010,Grzegorz Matecki |
Informatyka Teoretyczna On-line matching on bipartite graphs |
| We consider bipartite matching in the on-line version as follows. There is a bipartite graph G=(U,V,E), in which vertices in V are given a priori and each vertex u in U arrives in the on-line fashion (with all its incident edges). An on-line algorithm matches each such vertex u to a previously unmatched adjacent vertex in V, if there is one. Decisions made by the algorithm are irrevocable. The objective is to maximize the size of the resulting matching. It is easy to observe that any greedy algorithm (never leave vertex u unmatched if a match is possible) matches at least n/2 edges where n is the size of the optimal matching with G given at once. This number is optimal and there is no better algorithm. We propose the following modification of an on-line matching. The algorithm matches each incoming vertex u in U to a set S(u) of adjacent vertices in V (instead of one vertex). In case when S(u) and S(x) for already existing x in U are not disjoint the algorithm must remove all common vertices from S(x). Additionally, the algorithm has to obey the rule: each S(x) must not become empty if only it was initialized with a nonempty set of vertices. An algorithm satisfying the above condition is called adaptive. In this approach a vertex u in U can be always matched to a vertex from S(u) (if S(u) is not empty). Therefore, the number of matched edges is equal to the number of nonempty sets S(u). We are going to present the optimal adaptive on-line algorithm which breaks n/2 barrier and matches at least 0.589n+o(n) edges. |
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06.01.2010 Maria Chmaj |
Podstawy Informatyki Równość |
| Rozdział 8. książki Changa i Lee o automatycznym dowodzeniu twierdzeń. |
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17.12.2009 Przemysław Mazur |
Algorytmiczne Aspekty Kombinatoryki Hat guessing game on graphs |
| Suppose that n bears are standing on the n vertices of a graph G. Each of them can see only colleagues from the adjacent vertices. Suddenly, on every bear's head, a hat falls down in one of k available colors. Then, after a moment of looking around, each bear must write down the supposed color of its own hat (meanwhile they cannot communicate). The bears win the game if at least one of them correctly guesses the color of his hat. Otherwise they loose. The maximum number of hat colors for which the bears have a winning strategy on a graph G is called the bear number of G, denoted by mi(G). For instance, mi(C_4)=3, while mi(C_5)=2. We will present sveral results and conjectures on this fascinating game. |
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16.12.2009 Radosław Kożuch |
Informatyka Teoretyczna An O(m^2 n) Alghorithm for Minimum Cycle Bases of Graph |
| Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. Cycle bases are a compact description of the set of all cycles of a graph and cycle bases consisting of short cycles or, in weighted graphs, of small weight cycles are preferable. I will present an algorithm for computing minimum cycle basis in an undirected graph in time O(E^2*V + E*V^2*log V). References:Telikepalli Kavitha, Kurt Mehlhorn, Dimitrios Michail, Katarzyna E. Paluch, An O(m^2 n) Algorithm for Minimum Cycle Basis of Graphs, Algorithmica (2008) 52: 333–349 |
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16.12.2009 Jan Hązła |
Podstawy Informatyki Linear Resolution in FOL |
| Rozdział 7. książki Changa & Lee. |
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10.12.2009 Andrzej Grzesik |
Algorytmiczne Aspekty Kombinatoryki The chromatic sum of a graph |
| The "chromatic sum" of a graph is the smallest sum of colors among all proper colorings with natural numbers. The "strength" of a graph is the minimum number of colors necessary to obtain its chromatic sum. Some existing problems and results about these parameters will be presented. |
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09.12.2009 Bartłomiej Bosek, Tomasz Krawczyk |
Informatyka Teoretyczna Subexponential algorithm for on-line chain partitioning problem (cont.) |
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03.12.2009 Mateusz NikodemAGH University of Science and Technology |
Algorytmiczne Aspekty Kombinatoryki Foult-tolerant dominating sets |
| Assume that each vertex of a graph G is the possible location for an "intruder" such as a thief, or some possible processor fault in a computer network. Consider a set S which is a subset of V(G). Assume that any s of S is able to detect an intruder at any vertex in its closed neighborhood N[s] with identifying the location in N[s]. We want to construct the set S such that an intruder will be detected in any vertex of G, even if k vertices of S are liars and l vertices of S are false-alarm makers. Some necessary and and sufficient conditions for such set S will be presented. |
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02.12.2009 Grzegorz Matecki |
Informatyka Teoretyczna Golden ratio in on-line chain partitioning problem |
| Consider a game with two players. The first one, called Spoiler, reveals points of an order, one at a time. The second one, called Algoritm, assigns each new point to some chain and so maintains a chain partition of an incoming order. The aim of Algoritm is to use the smallest number of chains as possible. Whereas Spoiler tries to force Algoritm to use as much as possible different chains. The talk is on a variant of this game where Spoiler reveals a semi-order and each new point is maximal at a time it arrives. We present a strategy for Algoritm using at most gw chains where g is a golden ratio (g=1,618..) and w is optimal (off-line) number of chains. Moreover, we prove that it is best possible. Our strategy somehow induces a system of linear inequalities incorporating w (off-line optimum) as well as the number of chains used by Algoritm. The solution of this system shows how the golden ratio is involved in decisions made by Algoritm. References:Stefan Felsner, Kamil Kloch, Grzegorz Matecki and Piotr Micek, On-line chain partitioning of up-growing semi-orders, submitted |
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26.11.2009 Tomasz DzidoUG Gdańsk |
Algorytmiczne Aspekty Kombinatoryki Altitude of wheels, wheel-like graphs and some r-partite graphs |
| In my talk I want to present definition, properties and all known results for Altitude of different graphs. In addition, I want to show some new results. I will consider Altitude of wheels and wheel-like graphs as fans, gear graphs and helms. Finally, I will present some values and bounds for Altitude of 2 i 3-partite graphs. |
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25.11.2009 Andrei KrokhinDurham University |
Informatyka Teoretyczna On the hardness of losing weight |
| Local search algorithms iteratively improve solutions by checking whether it is possible to find a better solution in the local neighborhood of the current solution. The local neighborhood is usually defined as the set of solutions that can be obtained by one (or more generally, at most k for some fixed k) elementary changes. Large values of k can give better results; however, the brute force search of the local neighborhood is not feasible for larger k. Parameterized complexity gives a convenient framework for studying the question whether there is an efficient way of searching the local neighborhood. In the talk, I will briefly overview parameterized complexity, summarize recent results in this direction, and explain in more detail the analysis of the problem of finding minimum weight solutions for Boolean CSP. (Joint work with Dániel Marx) |
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25.11.2009 Piotr Faliszewski |
Podstawy Informatyki Distance Rationalizability of Voting Rules |
| A voting rule is an algorithm for determining the winner in an election. One can easily come up with many different voting rules, but it is also important to justify why a given rule is natural/useful. There are several approaches that have been used to justify the proposed rules. One justification is to show that a rule satisfies a set of desirable axioms that uniquely identify it. Another is to show that the calculation that it performs is actually maximum likelihood estimation relative to a certain model of noise that affects voters (MLE approach). The third approach, which has been recently actively investigated, is the so-called distance rationalizability framework. In it, a voting rule is defined via a class of consensus elections (i.e., a class of elections that have a clear winner) and a distance function. A candidate c is a winner of an election E if c wins in one of the consensus elections that are closest to E relative to the given distance. In this talk, we show that essentially any voting rule is distance-rationalizable if we do not restrict the two ingredients of the rule: the consensus class and the distance. Thus distance rationalizability of a rule does not by itself guarantee that the voting rule has any desirable properties. However, we demonstrate that the distance used to rationalize a given rule may provide useful information about this rule's behavior. Specifically, we identify a large class of distances, which we call votewise distances, and show that if a rule is rationalized via a distance from this class, many important properties of this rule can be easily expressed in terms of the underlying distance. This enables us to provide a new characterization of scoring rules and to establish a connection with the MLE framework. We also give bounds on the computational complexity of the winner determination problem for distance-rationalizable rules. |
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19.11.2009 Michał Farnik |
Algorytmiczne Aspekty Kombinatoryki Riemann-Roch versus chip-firing |
| A finite graph can be viewed as a discrete analogue of a Riemann surface, or smooth complete complex curve. During my talk I will present some recent results using this analogy in the context of linear equivalence of divisors. In particular I will formulate a graph-theoretic analogue of the classical Riemann-Roch Theorem and show how to apply it to characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph. |
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18.11.2009 04.11.2009 28.10.2009 Bartłomiej Bosek Tomasz Krawczyk |
Informatyka Teoretyczna Subexponential algorithm for on-line chain partitioning problem |
An on-line chain partitioning algorithm receives the elements of a poset, point by point, from some externally determined list. When a new point is presented the algorithm learns its comparability status with previously presented points and makes an irrevocable choice of a color, keeping the invariant that all points with the same color form a chain. The choice of a color is made without knowledge of future points. The number of colors used by an on-line algorithm is usually compared to the width w of the poset. Kierstead showed that there exists an on-line algorithm that covers any poset with (5^w-1)/4 chains. On the other hand Szemeredi proved that any on-line algorithm for the on-line chain partitioning problem has to use at least (w^2+w)/2 colors. We reduce the huge gap between the exponential upper bound and the polynomial lower bound by improving the upper bound to the subexponential function: in fact we show an on-line chain partitioning algorithm that uses at most w^O(log w) many colors. |
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18.11.2009 Sylwia Antoniuk |
Podstawy Informatyki Rezolucja |
| Rezolucja jako rewolucja, w relacji do rewelacji. |
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05.11.2009 Hao LiUniversite de Paris-Sud |
Algorytmiczne Aspekty Kombinatoryki Rainbow subgraphs in edge colored graphs |
| Jest to kolejny wykład w ramach inicjatywy SSAK. Czas - 16:30, miejsce - kampus AGH (budynek B7, sala 1.9), obecność - obowiązkowa. |
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04.11.2009 Rafał Pajdzik, UJ |
Podstawy Informatyki FOL |
| Chapters 3,4 on First Order Logic from `Symbolic Logic and Mechanical Theorem Proving' by Chin-Liang Chang and Richard Char-Thung Lee. |
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29.10.2009 Paweł ŻylińskiUG Gdańsk |
Algorytmiczne Aspekty Kombinatoryki Guarding grids and related graph problems |
| The guards problem in grids is one of the art gallery problems and it was formulated by Ntafos in 1986. A grid P is a connected union of vertical and horizontal line segments; a grid may be thought of as an orthogonal polygon with holes, with very thin corridors. A point x in P can see point y in P if the line segment xy is a subset of P. A set of points S, being a subset of P, is a guard set for grid P if any point of P is seen by at least one guard in S. During my talk, I will present several variants of the problem, including cooperative guards, fault-tolerant guards, mobile guards, and the pursuit evasion problem, and discuss their relation to the well-known graph theory problems, e.g., matching, coloring, domination. |
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28.10.2009 Marek Zaionc |
Podstawy Informatyki Schwichtenberg style lambda definability is undecidable |
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22.10.2009 Dominik Kwietniak |
Algorytmiczne Aspekty Kombinatoryki Combinatorial dynamics of one-dimensional maps |
| Problems connected to one-dimensional dynamics are often expressed in the language of combinatorics. To solve them one deals mainly with permutations, graphs, etc. During this talk, an introduction to the subject will be presented. If time permits, some open problems, in which combinatorial approach might provide a solution, will be included. |
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21.10.2009 14.10.2009,Piotr Micek,Bartosz Walczak |
Informatyka Teoretyczna Graph eating games |
| Alice and Bob share a connected graph. Its vertices are weighted with non-negative values summing up to one. The players eat the vertices alternately one by one (starting with Alice) until no vertex is left. The rule they have to obey is that after each move the vertices eaten so far form a connected subgraph of the original graph. Both players want to maximize their final gain, i.e., the total weight of the vertices they have eaten. This game for a cycle is known as the pizza eating problem. Recently, Knauer, Micek and Ueckerdt proved that Alice can eat 4/9 of any cycle (pizza), which is best possible and settles the conjecture of Peter Winkler. In the general game, Alice cannot guarantee herself any positive constant gain on all connected graphs. Curiously, the parity of the number of vertices makes a difference. Examples of graphs with small Alice's gain having an odd number of vertices need a very rich structure, contrary to strikingly simple examples with an even number of vertices. In particular, there are trees with an even number of vertices which are very bad for Alice, while she can guarantee herself a positive constant gain on all odd trees. We wish to introduce the audience to this and similar games on graphs. Our techniques are quite general and seem to be applicable to other combinatorial games as well. |
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21.10.2009 Mareusz Drewienkowski, UJ |
Podstawy Informatyki Gry w wielomiany (Playing polynomials) |
| We will hear about the process of reconstruction of polynomials from the finite number of examples. The talk is based on the 1997 paper by J. Małolepszy, M. Moczurad and M. Zaionc entitled "Schwichtenberg style lambda definability is undecidable" published in Lecture Notes in Computer Science 1210, pp. 267-283. |
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15.10.2009 Piotr Cieślik |
Algorytmiczne Aspekty Kombinatoryki Set intersection, perfect graphs, and voting in agreeable societies |
| When is agreement possible? An important aspect of group decision-making is the question of how a group makes a choice when individual preferences may differ. Clearly, when making a single group choice, people cannot all have their "ideal" preferences, i.e, the options that they most desire, if those ideal preferences are different. However, for the sake of agreement, people may be willing to accept as a group choice an option that is merely "close" to their ideal preferences. |
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14.10.2009 Marek Zaionc |
Podstawy Informatyki Two open problems on lambda definability |
| We discus two open lambda definability problems. 1. It can be shown that any lambda definable operation on numbers can be synthesize from the finite number of examples. We will discuss the same problem for lambda definable word operations. 2. We address the problem of lambda definable tree operations. Is it true that the set of all lambda definable tree operations is NOT finitely generated. |
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07.10.2009 Tomasz Jurkiewicz |
Informatyka Teoretyczna Breaking through the O(m^2 n) Barrier for Minimum Cycle Bases |
| Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. Cycle bases are a compact description of the set of all cycles of a graph and cycle bases consisting of short cycles or, in weighted graphs, of small weight cycles are preferable. We will present an algorithm for computing general minimum weight cycle bases in time O(E^omega) for general graphs; here V and E denote the number of nodes and edges, respectively, and omega is the exponent of the fastest matrix multiplication algorithm. Our algorithm is the first to run faster than Otilde(E^2 V). A key to our improved running time is the insight that the search for the minimum basis can be restricted to a set of candidate cycles of total length O(V E). References:Edoardo Amaldi, Claudio Iuliano, Tomasz Jurkiewicz, Kurt Mehlhorn and Romeo Rizzi. Breaking the O(m^2 n) Barrier for Minimum Cycle Basis, ESA 2009 |
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16.09.2009 Lê Đại Trí MẫnUniversity of Toronto |
Informatyka Teoretyczna An introduction to generalized combined traces |
| Mazurkiewicz traces were introduced by A. Mazurkiewicz in 1977 as a language representation of partial orders to model "true concurrency". The theory of Mazurkiewicz traces has been utilized to tackle not only various aspects of concurrency theory but also problems from other areas, including combinatorics, graph theory, algebra, and logic.
However, neither Mazurkiewicz traces nor partial orders can effectively model more complex relationships, e.g., "not later than". In this talk, I will introduce the theory of generalized combined traces (generalized comtraces). Generalized comtraces are generalizations of Mazurkiewicz traces, where generalized stratified order structures are used instead of partial orders. This allows us to model even the most general case of concurrent behaviors under the assumption that observations are stratified orders. The theory of generalized comtraces also provides us a wide variety of new and interesting problems to work on. References:R. Janicki and D.T.M. Le, Modelling Concurrency with Comtraces and Generalized Comtraces, preprint can be found at http://arxiv.org/abs/0907.1722 |
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10.06.2009 Michal Handzlik |
Informatyka Teoretyczna Online unit clustering |
| Online unit clustering is a clustering problem where classification of points is done in an online fashion, but the exact location of clusters can be modified dynamically. We study several variants and generalizations of online unit clustering problem, which are inspired by variants of packing and scheduling problems in the literature |
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03.06.2009 Michał WronaWrocław University |
Informatyka Teoretyczna Quantified Positive Temporal Constraints |
| A quantified constraint satisfaction problem (qcsp) is a version of a constraint satisfaction problem (csp) where variables occurring in an input formula can be not only existentially but also universally quantified. We say that a relation is temporal positive if it has a positive first order definition over the order of rational numbers. Our main contribution is a complexity characterization of qcsp(L) for all finite sets of positive temporal relations L. The complexity of these problems varies. Some of them are in LOGSPACE, some are NLOGSPACE-complete, P-complete, NP-complete, or PSPACE-complete. |
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28.05.2009 Andrzej RucińskiUAM Poznań |
Algorytmiczne Aspekty Kombinatoryki 2nd Discrete Integration Meeting & SSAK |
| Kampus AGH, budynek B7, sala 1.9, godzina 16:00. |
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28.05.2009 Andrzej RucińskiUAM Poznań |
Informatyka Teoretyczna 2nd Discrete Integration Meeting & SSAK |
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27.05.2009 Paweł Waszkiewicz |
Podstawy Informatyki T^omega as a universal domain |
| Following G. Plotkin [T^omega as a universal domain, J. of Comp. and System Sci. 17, 1978] We introduce a domain T^omega and a language LAMBDA, and show that LAMBDA-definability coincide with computability. Furthermore, we show that T^omega is universal, in the sense that every bounded-complete dcpo embeds in it. Finally, we demonstrate that every second-countable T_0 space topologically embeds in T^omega as an isochordal subspace. |
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20.05.2009 Sylwia Antoniuk |
Informatyka Teoretyczna Efficient on-line repetition detection |
| A repetition is a nonempty string of the form X^q, where q >= 2. Given a string S character by character and the value of q, the on-line repetition detection problem is to detect and report the first repetition in S, if it exists, in an on-line manner. Leung, Peng and Ting first studied the problem for q=2 and gave an O(m log^2 m) time algorithm, where m is the ending position of the first repetition in S. We improve the above work by reducing the time complexity to O(m log B), where B is the number of distinct characters in the first m characters of S. Moreover, we also solve the problem for q >= 3 with the same time complexity. |
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14.05.2009 Jan Jeżabek |
Algorytmiczne Aspekty Kombinatoryki Collecting weighted items from a dynamic queue |
| We consider the following problem: at each step some new weighted items are added to a dynamic queue (at any position) and some items from the front of the queue expire. We may collect one item at each step. Our objective is to maximize the total weight of collected items. For the analysis of this online problem we use the competitive ratio. It can be easily shown that the best possible competitive ratio is between (1 + sqrt(5)) / 2 = 1.618... and 2. We show better lower (1.632...) and upper (1.897...) bounds. References:Bieńkowski et al., Collecting Weighted Items from a Dynamic Queue, SODA '09 |
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13.05.2009 Andrzej Kukier |
Informatyka Teoretyczna Similarity Search in High Dimensions via Hashing |
| The nearest- or near-neighor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search/index structures for performing similarity search over high-dimensional data, e.g. image databases, document collections, time-series databases and genome databases. Unfortunately, all known techniques for solving this problem fall prey of the "curse of dimensionality". That is, the data structures scale poorly with data dimensionality: in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related data structures involves the inspection of a large fraction of the database, therby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determinig an approximate nearest neighbor should suffice for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search bases on hashing. The basic idea is to hash the points for the database so as to ensure that the probability of a collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives significant improvement in running time over other methods for searching in high-dimensional spaces based on hierachical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). |
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06.05.2009 Piotr Cieślik |
Informatyka Teoretyczna Edge-Coloring Bipartite Multigraphs in O(E log D) Time |
| For V, E and D denoting the cardinality of the vertex set, the cardinality of the edge set, and the maximum degree of a bipartite multigraph G, it is shown that a minimal edge-coloring of G can be computed in O(E log D) time. This result follows from an algorithm for finding a matching in a regular bipartite graph in O(E) time. |
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29.04.2009 Bartłomiej Bosek |
Informatyka Teoretyczna Posets omitting two incomparable chains of the same height |
| We consider a problem of partitioning a poset into chains by First-Fit algorithm. In general this algorithm uses arbitrarily many chains on a class of bounded width posets. In this paper we prove that First-Fit uses at most $4tw^2$ chains to partition any poset of width $w$ which does not induce two incomparable chains of height $t$. In this way we get a wide class of posets with polynomial bound for the on-line chain partitioning problem. We discuss also some consequences of our result for coloring graphs by First-Fit. References:B. Bosek, T. Kraczyk and E. Szczypka, First-Fit algorithm for on-line chain partitioning problem, manuscript, 2009. |
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23.04.2009 Paweł Prałat |
Algorytmiczne Aspekty Kombinatoryki Cleaning Regular Graphs with Brushes and Brooms |
| A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean d-regular graphs in this model, focusing on the asymptotic number for random d-regular graphs. We use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method to find the (asymptotic) number of brushes needed to clean a random d-regular graph using this algorithm (for fixed d). We further show that for any d-regular graph on n vertices at most n(d+1)/4 brushes suffice, and prove that for fixed large d, the minimum number of brushes needed to clean a random d-regular graph on n vertices is asymptotically almost surely n/4(d+o(d)). |
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16.04.2009 Bartłomiej Bosek i Tomasz Krawczyk |
Algorytmiczne Aspekty Kombinatoryki Coloring numbers of co-comparability graphs |
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02.04.2009 Torsten Ueckerdt |
Algorytmiczne Aspekty Kombinatoryki On Large Quadrant-Depth |
| We consider a point set P of n points in the plane with no two points sharing the same x or y-coord. For any (a,b) in [0,1]x[0,1] a point p in P is called (a,b)-deep if there are two opposite quadrants Q1 and Q2 of p such that Q1 \cap P >= a*n and Q2 \ cap P >= b*n. Moreover the pair (a,b) is called feasible if every finite point set has an (a,b)-deep point. In this talk we present the set F of all feasible pairs (a,b). |
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01.04.2009 Kolja Knauer |
Informatyka Teoretyczna Chip-Firing, Antimatroids, and Polyhedra |
| Starting from the chip-firing game of Björner and Lovász we consider a generalization to vector addition systems that still admit algebraic structures as sandpile group or sandpile monoid. Every such vector addition language yields an antimatroid. We show that conversely every antimatroid can be represented this way. The inclusion order on the feasible sets of an antimatroid is an upper locally distributive lattice. We characterize polyhedra, which carry an upper locally distributive structure and show that they can be modeled by chip-firing games with gains and losses. At the end we point out a connection to a membership problem discussed by Korte and Lovász. |
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26.03.2009 Stefan Felsner |
Algorytmiczne Aspekty Kombinatoryki Sampling from distributive lattices - the Markov chain approach |
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25.03.2009 Apoloniusz Tyszka |
Informatyka Teoretyczna A hypothetical upper bound for solutions of a Diophantine equation with a finite number of solutions |
Let E_n be the set of all equations of the form |
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25.03.2009 11.03.2009,Paweł Waszkiewicz |
Podstawy Informatyki A Lazy Introduction to Goedel's Theorems |
| See http://www.logicmatters.net/ for more info. |
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19.03.2009 Edward Szczypka |
Algorytmiczne Aspekty Kombinatoryki Thue games and the Lovasz local lemma |
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18.03.2009 04.03.2009 Grzegorz Gutowski |
Informatyka Teoretyczna Multicommodity Max-Flow Min-Cut Theorems |
The results of two papers will be presented: T. Leighton, S. Rao "Multicommodity Max-Flow Min-Cut Theorems and Their Use in Designing Approximation Algorithms" Abstract: In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities.
O. Günlük "A New Min-Cut Max-Flow Ratio for Multicommodity Flows" Abstract: We present an improved bound on the min-cut max-flow ratio for multicommodity flow problems with specified demands. To obtain the numerator of this ratio, capacity of a cut is scaled by the demand that has to cross the cut. In the denominator, the maximum concurrent flow value is used. Out new bound is proportional to log(k*) where k* is the cardinality of the minimal vertex cover of the demand graph. |
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05.03.2009 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Chips on graphs; five easy pieces |
| Referat odbędzie się w sali 0089. |
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26.02.2009 Maciej Ulas |
Algorytmiczne Aspekty Kombinatoryki Experimental mathematics in action |
| Wykład odbędzie się w sali 0089. |
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28.01.2009 Rafał Józefowicz |
Informatyka Teoretyczna On-line interval coloring with packing constraints |
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21.01.2009 Lech Duraj |
Informatyka Teoretyczna Optimal orientation problem for different graph classes |
| We consider the problem of giving direction to every edge of an undirected graph, such that the number of connected pairs of vertices is maximal. In an on-line variant, the vertices of graph are given one-by-one, and the algorithm's decisions are permanent. Despite the fact that off-line algorithm always gives an orientation with O(n^2) connected pairs, the optimal outcome of the on-line algorithm can vary from O(n) to O(n^2), depending of the additional rules imposed on the graph. I'll present several possible sets of rules with different strategies and outcomes. |
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21.01.2009 Mateusz Kostanek |
Podstawy Informatyki Regular Languages and Stone Duality |
| We give a new account of the relationships among varieties of regular languages, varieties of finite semigroups, and their characterization in terms of "implicit identities." Our development, which is essentially topological in character, is based on the duality (established by Stone) between Boolean algebras and certain topological spaces (which are now called "Stone spaces"). This duality does not seem to have been recognized in the literature on regular languages, even though it is well known that the regular languages over a fixed alphabet form a Boolean algebra and that the "implicit operations" with a fixed number of operands form a Stone space. Na podstawie artykulu: N. Pippenger, Regular Languages and Stone Duality, Theory Comput. Systems30, 121-134 (1997). |
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14.01.2009 Michał MorayneTU Wrocław |
Informatyka Teoretyczna How to choose the best twins |
| We consider a variant of the secretary problem where each candidate has an identical twin that applies for the same job. We find both an optimal strategy how to choose one of the best twins and the probability of success as well as the assymptotics for this probability. (with the number of candidates tending to infinity). |
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07.01.2009 Bartosz Walczak |
Informatyka Teoretyczna Fast route planning in road networks |
| I will discuss the problem of finding an optimal route between two specified nodes in a transportation network (represented as a weighted graph). When the graph is very big (as real road networks are) and there are lots of queries, one cannot afford to run a simple Dijkstra search for each individual query. Some additional structure is necessary in order to be able to answer the queries efficiently. A natural idea is to exploit hierarchical structure of road networks: only "important" roads are worth considering far away from the source and destination. Several commercial route planning systems use the formal hierarchy (freeways, national highways etc.) for this purpose. However, formal hierarchies are not perfect, and the route found this way not be optimal. The modern approach is to compute an appropriate hierarchy from scratch in the preprocessing step, based on the bare graph. After that, each query can be quickly answered with an exact optimal solution. I will present several ways of achieving this goal. References:P. Sanders, D. Schultes, Engineering Fast Route Planning Algorithms, 2007. P. Sanders, D. Schultes, Engineering Highway Hierarchies, 2006. |
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07.01.2009 Michał Pokrywka |
Podstawy Informatyki Algorytmy kolejkowania danych w sieciach komputerowych |
| cd z poprzedniego roku. |
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17.12.2008 Sebastian Czerwiński University of Zielona Góra |
Informatyka Teoretyczna Short proof of Combinatorial Nullstellensatz |
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17.12.2008 Michał Pokrywka |
Podstawy Informatyki Algorytmy kolejkowania danych w sieciach komputerowych |
| Przedstawię dwie publikacje godne uwagi opisujące modele matematyczne dwóch rozwiązań. Jedna dotyczy algorytmu RED (Random Early Detection) a druga HFSC (Hierarchical Fair Service Curve). Algorytm RED pozwala w sieciach wysokich prędkości na zapobieganie zdominowaniu pasma przez jeden strumień danych za pomocą losowego odrzucania pakietów (z prawdopodobieństwem ważonym). Algorytm HFSC pozwala dla kolejki danych zdefiniować parametry ,,krzywych usług'' (ang. service curves) do dynamicznego sprawiedliwego podziału pasma, zachowując parametry strumieni real-time. |
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11.12.2008 Bartosz Walczak |
Algorytmiczne Aspekty Kombinatoryki Schnyder trees and grid embeddings of planar graphs |
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03.12.2008 Libor Barto Charles University, Prague |
Informatyka Teoretyczna Constraint Satisfaction Problems of Bounded Width |
| We prove an algebraic characterization of applicability of the bounded width algorithm solving problem posted by Larose and Zadori. |
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26.11.2008 Jan Jeżabek |
Informatyka Teoretyczna Resource Augmentation for Packet Switching with Agreeable Deadlines |
| We study a scheduling problem known as on-line packet switching. We utilize a technique called resource augmentation, where an optimal off-line algorithm is compared against an on-line algorithm with more processing power, i.e. one that can transmit more than one packet per unit of time. A previous result showed that regardless of the processing power of the on-line algorithm there are instances on which it is outperformed by the off-line algorithm. We will show that if the jobs in the instance have aggreeable deadlines (i.e. for any jobs i,j the time interval where i is available is not contained in the interior of the time interval where j is available) a resource augmented on-line algorithm which executes two jobs per time slot will always perform at least as well as the optimal off-line algorithm. |
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20.11.2008 Bartłomiej Bosek |
Algorytmiczne Aspekty Kombinatoryki First fit algorithm for on-line chain partitioning problem |
| On-line chain partititoning of orders can be viewed as the game between two-person between: Algorithm and Spoiler. The game is played in rounds. During each round Spoiler introduces a new point of an order with its comparability status to previously presented points while Algorithm gives it a color in such a way that the points with the same color form a chain. We consider the First Fit Algorithm (FFA) - it gives the first possible color to each introduced point. There is a relatively easy strategy for a Spoiler that forces FFA to use arbitrary many colors even on orders of width 2. We proved that if the game is played on orders of width w that do not contain k+k (two incomparable chains of height k) as subposet then FFA uses no more than 4kw^2 colors. Joint work with Tomasz Krawczyk and Edward Szczypka. |
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19.11.2008 12.11.2008,Michał Staromiejski |
Informatyka Teoretyczna Computing isomorphisms between certain finite rings |
| For a given finite field $F$ and two polynomials $f, g \in F[X]$, there is a polynomial-time algorithm deciding whether the (finite) rings $F[X]/(f)$ and $F[X]/(g)$ are isomorphic? However, a question about constructing an isomorphism provided the rings are isomorphic seems to be more challenging. In 1991, Lenstra showed that such an isomorphism can be computed in deterministic polynomial time if the polynomials $f$ and $g$ are irreducible over $F$. There is an obvious algorithm that can solve the general problem if we allow randomization. In the talk I will present partial results which may help to find a deterministic polynomial-time algorithm. |
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05.11.2008 29.10.2008,Piotr Micek |
Informatyka Teoretyczna How to eat 4/9 of a pizza |
| Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously-taken, we provide a strategy for the starting player to get $\frac{4}{9}$ of the pizza. This is best possible and settles a conjecture of Peter Winkler. References:Kolja Knauer, Piotr Micek and Torsten Ueckerdt, How to eat 4/9 of a pizza, manuscript |
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30.10.2008 Piotr Micek |
Algorytmiczne Aspekty Kombinatoryki Efficient graph packing via game coloring |
| Kierstead and Kostochka's abstract: The game coloring number gcol(G) of a graph G is the least k such that if two players take turns choosing the vertices of a graph then either of them can insure that every vertex has less than k neighbors chosen before it, regardless of what choices the other player makes. Clearly, gcol(G)<= DELTA(G)+1. Sauer and Spencer proved that if two graphs G_1 and G_2 on n vertices satisfy 2*DELTA(G_1)*DELTA(G_2)<n then they pack, i.e., there is an embedding of G_1 into the complement of G_2. we improve this by showing that if (gcol(G_1)-1)*DELTA(G_2)+(gcol(G_2)-1)*DELTA(G_1)<n then G_1 and G_2 pack. To our knowledge this is the first application of coloring games to a non-game problem. References:H.A. Kierstead, A.V. Kostochka, Efficient graph packing via game coloring |
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15.10.2008 08.10.2008,Piotr Micek, Bartosz Walczak |
Informatyka Teoretyczna Summer conferences 2008 |
| Selected results and open problems from summer conferences are presented |
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15.10.2008 Jarosław Duda |
Podstawy Informatyki Asymmetric numeral systems |
| We will speak about new approaches to entropy encoding. We present various generalizations of standard numeral systems which are optimal for encoding sequences of equiprobable symbols as asymmetric numeral systems - optimal for freely chosen probability distributions of symbols. It has some similarities to Range Coding but instead of encoding a symbol in choosing a range, we spread these ranges uniformly over whole interval. This leads to a simpler en- coder - instead of using two states to define range, we need only one. This approach is truly universal - we can get from extremely precise encoding (ABS) to extremely fast with possibility to additionally encrypt the data (ANS). This encryption uses a key to initialize a random number generator, which is used to calculate the coding tables. Such preinitialized encryption has an additional advantage: it's resistant to brute force attack - in order to check a key we have to make the whole initialization. We will also show that using ANS we can get an error correction method which is resistant to pessimistic cases. |
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09.10.2008 Wiktor Żelazny |
Algorytmiczne Aspekty Kombinatoryki Lucky labelings of graphs |
| Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by L(G). Using algebraic methods we prove that L(G)≤k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that L(T)≤2 for every tree T, and L(G)≤3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that L(G)≤100 280245065 for every planar graph G. Nevertheless we offer a provocative conjecture that L(G)≤Chi(G) for every graph G. |
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12.06.2008 Lech Duraj |
Algorytmiczne Aspekty Kombinatoryki Interval chromatic number of planar graphs |
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11.06.2008 Zbigniew LoncPolitechnika Warszawska |
Informatyka Teoretyczna Small transversals in hypergraphs |
| Zbiór wierzchołków hipergrafu, który przecina wszystkie jego krawędzie nazywamy transwersalem. Klasyczny algorytm zachłanny znajdujący "mały" transwersal w hipergrafie wybiera w każdym kroku do transwersala wierzchołek należący do największej liczby krawędzi nie zawierających wierzchołków już wybranych. Analiza tego algorytmu (autorstwa Chvátala i McDiarmida) prowadzi do pewnych górnych ograniczeń na najmniejszą liczność transwersala w jednorodnym hipergrafie o zadanej liczbie wierzchołków i krawędzi. Referat będzie poświęcony pewnej modyfikacji tego algorytmu zachłannego. Jego analiza prowadzi do poprawienia ograniczeń Chvátala i McDiarmida. Rezultaty te wiążą się ze znanym kombinatorycznym problemem wyznaczenia tzw. liczb Turána. W szczególności implikują nowe dolne ograniczenia na liczby Turána w pewnych szczególnych przypadkach. |
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04.06.2008 Oleg PikhurkoCarnegie Mellon University |
Informatyka Teoretyczna The Stability Method for the Hypergraph Turán Problem |
| For a k-graph F, the Turan function ex(n,F) is the maximum size of a k-graph on n vertices not containing a copy of F. Although this function was introduced by Turan yet in 1941, very few non-trivial cases have been solved and there is an abundace of open problems. We survey some recent results, concentrating on the so-called stability approach that was used to obtain some of them. |
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29.05.2008 Artur Szymański (AGH) |
Algorytmiczne Aspekty Kombinatoryki Strive for Photorealism - A Brief History of 3D Computer Graphics |
| This will be a survey talk presenting a development of 3D computer graphics from its birth at MIT and UU in late 60' to present day. During this journey through time, we will outline a number of algorithms for shading and rendering (ray tracing, radiosity, photon mapping), as well as their application and influence on film industry. |
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28.05.2008 Leszek Horwath |
Informatyka Teoretyczna Simple wildcard matching |
| Brief review over wildcar matching algorithms. Introducting the new deterministic method of O(nlgm) complexity using Fast Fourier Transformation. |
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28.05.2008 Ken-etsu Fujita, Shimane Univ. Japan |
Podstawy Informatyki A translation from lambda2 into lambda_exists |
| This talk shows that there exist translations between polymorphic lambda calculus and a subsystem of minimal logic with existential types, which form a Galois insertion (embedding). The translation from polymorphic lambda calculus into the existential type system is the so-called call-by-name CPS-translation that can be expounded as an adjoint from the neat connection. The construction of an inverse translation is investigated from a viewpoint of residuated mappings. The duality appears not only in the reduction relations but also in the proof structures such as paths between the source and the target calculi. From a programming point of view, this result means that abstract data types can interpret polymorphic functions under the CPS-translation. We may regard abstract data types as a dual notion of polymorphic functions. This talk is based on an extended and improved version of the paper presented at TLCA2005. |
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21.05.2008 Jarosław Duda |
Informatyka Teoretyczna Combinatorial invariants for graph isomorphism problem |
| Some topological invariants for finite graphs that can be calculated in a polynomial time are presented. They may be useful in recovering the graph up to isomorphism. At least we will see how much information they do code. |
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14.05.2008 07.05.2008 Przemysław Broniek |
Informatyka Teoretyczna Computational complexity of solving equation systems |
We consider the computational complexity of determining whether a system of equations over fixed algebra A has a solution. This leads to two problems, SysTermSat(A) and SysPolSat(A), in which equations are built out of terms or polynomials, respectively. We are interested in characterizing those algebras, for which SysPolSat can be solved in a polynomial time. The problem has been widely studied and is open in general. We prove that the Constraint Satisfaction Problem for relational structures is polynomially equivalent to SysTermSat over unary algebras. This gives that Dichotomy Conjecture for CSP is equivalent to Dichotomy Conjecture for SysTermSat over unary algebras. We also give other partial characterizations of computational complexity of SysTermSat(A), e.g. for algebras with generic operations taking only few values. This covers wide class of four-element unary algebras. |
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07.05.2008 Bożena Woźna, Akademia im. Jana Długosza, Częstochowa |
Podstawy Informatyki Ograniczona Weryfikacja Modelowa dla systemów wieloagentowych i systemów z czasem |
| W referacie przedstawię wyniki moich badań w zakresie automatycznej weryfikacji modelowej systemów czasu rzeczywistego oraz systemów wieloagentowych. Wyniki te zostały osiągnięte we współpracy z prof. Wojciechem Penczkiem (IPI PAN Warszawa), dr Andrzejem Zbrzeznym (AJD, Częstochowa) oraz dr Alessio Lomuscio (UCL, Londyn). W szczególności opowiem o zaproponowanej przeze mnie Ograniczonej Weryfikacji Modelowej, którą zastosowałam zarówno do systemów z czasem jak i systemów wieloagentowych. Zrobię również krótkie wprowadzenie do formalizmów stosowanych w automatycznej weryfikacji modelowej wy?ej wymienionych systemów. System czasu rzeczywistego to (zgodnie z definicją IEEE) system, którego poprawność działania zależy nie tylko od poprawności logicznych rezultatów, lecz również od czasu, w jakim te rezultaty są osiągane. Systemy czasu rzeczywistego znajdują zastosowanie między innymi w przemyśle do nadzorowania procesów technologicznych, przy implementacji protokołów komunikacyjnych, w planowaniu i kontroli ruchu lotniczego, itd. Agent to jednostka, która działa w pewnym ustalonym środowisku, jest zdolna do komunikowania się, monitorowania swego otoczenia i podejmowania autonomicznych decyzji. System wieloagentowy to sieć komunikujących się i współpracujących między sobą agentów, realizujących zarówno wspólne jak i prywatne cele. Systemy wieloagentowe mają już swoją ugruntowaną pozycję w wielu dziedzinach związanych z technologią informacji, np.: w inżynierii oprogramowania, e-handlu, sieciach telekomunikacyjnych, automatycznym wnioskowaniu i argumentacji, wspomaganiu zarządzaniem w przedsiębiorstwie, itd. Weryfikacja modelowa jest jedną z najbardziej rozpowszechnionych metod automatycznej weryfikacji poprawności systemów czasu rzeczywistego oraz systemów wieloagentowych. Pierwsze prace na ten temat ukazały się w 1981 roku i od tamtego czasu trwa nieustanny rozwój narzędzi wykorzystujących udoskonalane algorytmy. Różnorodność dostępnych podejść, jak też rozwiązań, jest wynikiem istnienia wielu modeli dla wyżej wymienionych systemów, np. przeplotowych i nieprzeplotowych, jak też wielu metod opisu własności tych systemów, np. poprzez automaty, algebry procesów lub logiki temporalne. Istotny postp w dziedzinie weryfikacji dokonał się w 1990 roku po opracowaniu metod bazujących na obliczeniach symbolicznych, wykorzystujących formalizm Boolowskich Diagramów Decyzyjnych. Następny krok do przodu został wykonany w 1999 roku po sprowadzeniu problemu weryfikacji modelowej do problemu testowania spełnialności dla formuł zdaniowych i wykorzystaniu efektywnych algorytmów dla tego ostatniego problemu. |
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30.04.2008 Jan Hązła |
Informatyka Teoretyczna Simplified O(n) planarity algorithm |
| I will present O(n)-time methods for planar embedding and Kuratowski subgraph isolation that were inspired by the Booth-Lueker PQ-tree implementation of the Lempel-Even-Cederbaum vertex addition method. Instead of performing comprehensive tests of planarity conditions embedding the edges from a vertex to its descendants, we take the edge to be the fundamental unit of addition to the partial embedding while preserving planarity. This eliminates the batch planarity condition testing in favor of a few localized decisions of a path traversal process, and it exploits the fact that subgraphs can become biconnected by adding a single edge. The method is presented using only graph constructs, but the definition of external activity, path traversal process and theoretical analysis of correctness can be applied to optimize the Shih-Hsu PC-tree as well. References:John M. Boyer, Wendy J. Myrvold, On the Cutting Edge: Simplified O(n) Planarity by Edge Addition , Journal of Graph Algorithms and Applications 8 (3), 241-273 (2004) |
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23.04.2008 Sebastian CzerwińskiUniversity of Zielona Gora |
Informatyka Teoretyczna On a conjecture of Brown, Graham, and Landman |
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23.04.2008 Mateusz Juda |
Podstawy Informatyki Arytmetyka Heytinga |
| Na podstawie skryptu Thomasa Streichera (patrz poprzednie referaty tego samego autora). |
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16.04.2008 Maria Chmaj |
Informatyka Teoretyczna A linear-time algorithm for finding dominators in a flowgraph |
| The problem of finding dominators in a flowgraph arises in many kinds of global code optimization and other settings. In 1979 Lengauer and Tarjan gave an almost-linear-time algorithm to find dominators. Several attempts at a linear-time algorithm were unsuccessful. I will talk about a linear-time algorithm which Georgiadis and Tarjan gave in 2004. References:L.Georgiadis, R.Tarjan, Finding dominators revisted, In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), 862-87 |
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16.04.2008 Tomasz Połacik, Uniw. Śląski |
Podstawy Informatyki Modele Kripkego dla teorii pierwszego rzędu |
| Druga część referatu jest poświęcona modelom Kripkego dla logiki pierwszego rzędu. Omówimy podstawowe własności modeli i twierdzenie o pełności. Przedstawimy w niej również rezultaty dotyczące konstrukcji modeli Kripkego dla intuicjonistycznych teorii pierwszego rzędu, ze szczególnym uwzglednieniem Arytmetki Heytinga. |
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09.04.2008 Tomasz Połacik, Uniw. Śląski |
Podstawy Informatyki Modele Kripkego dla intuicjonistycznej logiki zdań |
| W pierwszej części referatu zajmiemy się zagadnieniami związanymi z modelami Kripkego dla intuicjonistycznej logiki zdań. Poza podstawowymi własnościami i intuicjami dotyczącymi modeli, przedstawione zostanie twierdzenie o pełności Rachunku Heytinga względem semantyki kripkowskiej. Omówimy też najważniejsze konstrukcje modeli Kripkego i pokażemy semantyczne dowody niektórych własności Rachunku Heytinga. |
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02.04.2008 Paweł Waszkiewicz |
Podstawy Informatyki O analizie infinitezymalnej w światach gładkich |
| Zanim zaproponowano pojęcie granicy, matematycy tacy jak Fermat czy Leibnitz posługiwali się w dowodach swoich twierdzeń analitycznych pojęciem wartości nieskończenie małych. Podczas seminarium opowiem o rachunku wartości nieskończenie małych, który przypomina metody analityczne sprzed 350-400 lat, ale - w odróżnieniu od nich - jest doskonale precyzyjny. Modele tego rachunku nz. światami gładkimi. Co czyni światy gładkie ciekawymi jest m.in. fakt, iż w takim świecie prosta rzeczywista R jest prawdziwym continuum, tzn. nie można z niej wydzielić żadnego nietrywialnego podzbioru. (Podzbiór U wydziela się z R, jeśli dla każdego x z R albo x należy do U, albo x nie należy do U.) Wykład przygotowałem na podstawie: J.L.Bell, A Primer on Infinitesimal Analysis, Cambridge Univ. Press, 1998. Google: ,,John L. Bell''. Warto. |
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26.03.2008 Karol Kosinski |
Informatyka Teoretyczna Faster algorithms for finding lowest common ancestors in directed acyclic graphs |
| We are given two new methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on n vertices and m edges. The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag in time O(n*m). The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. The running time of the algorithm is O(n^2,575), so it improves the previously known O(n^2,688) time-bound for the general all-pairs LCA problem in dags by Bender, Pemmasani, Skiena and Sumazin. |
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19.03.2008 Jaroslav NesetrilCharles University, Prague |
Informatyka Teoretyczna Dualities for structures |
| Homomorphisms dualities are related to logic, model theory, partially ordered sets and of course to coloring of graphs. In this lecture we survey the recent development related to this notion. |
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19.03.2008 Mateusz Juda |
Podstawy Informatyki Introduction to Constructive Logic and Mathematics (II) |
| Omawiamy kolejne tematy ze skryptu Thomasa Streichera. Dzisiaj: - Constructive Arithmetic and Analysis - Constructive Real Numbers |
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13.03.2008 Zofia Miechowicz (Zielona Góra) |
Algorytmiczne Aspekty Kombinatoryki Rota's basis conjecture |
| Suppose that each edge of the complete bipartite graph K_n,n is assigned arbitrarily a basis of the Euclidean n-dimensional vector space V. Is it true that we may choose one vector for each edge from its basis so that the vectors lying around each vertex formed a basis of V? The problem is due to Gian-Carlo Rota, a positive answer would be a far reaching strengthening of the famous Dinitz problem for latin squares. We will report on progress, modifications and intriguing connections of this fascinating conjecture. |
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12.03.2008 Tomasz BartnickiUniversity of Zielona Gora |
Informatyka Teoretyczna Graph coloring with an uncooperative partner |
| Jacek and Placek color the vertices of a graph G alternately using given set of colors C, and with Jacek going first. Placek agrees to cooperate with Jacek by respecting the rule of a proper coloring. However, for some reason he does not want the job to be completed - his secret aim is to achieve a bad partial coloring. Is it possible for Jacek to complete the coloring somehow, in spite of Placek's insidious plan? If not, then how many additional colors are needed to guarantee that the graph can be successfully colored, no matter how clever Placek is?
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12.03.2008 Jakub Kozik |
Podstawy Informatyki Jak wiele formuł ma konstruktywne dowody? |
| W referacie przedstawię wyniki otrzymane z A. Genitrini dotyczące ilościowego porównania zdaniowych logik: klasycznej i intuicjonistycznej. W rozważanych formułach dopuszczamy wszystkie zwykle używane łączniki (koniunkcja, alternatywa, implikacja) oraz stałą "absurd" (bottom). Nasz główny wynik można nieformalnie wypowiedzieć jako: "około 5/8 tautologii logiki klasycznej ma konstruktywne dowody." Ciekawym wynikiem dodatkowym jest zgodność dwóch, pozornie niezależnych, metod liczenia gęstości. |
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06.03.2008 Jan Jeżabek |
Algorytmiczne Aspekty Kombinatoryki Four colors suffice to distinguish neighboors by multisets |
| A proof of the following result will be presented: every connected graph with more than one edge has a 4-coloring of the edges such that no two adjacent vertices get the same multisets of colors. It is conjectured that the best possible constant is 3, even if colors are positive integers 1,2,3, and we wish to distinguish neighboors by sums rather than just multisets. |
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05.03.2008 Maciej Chociej |
Informatyka Teoretyczna Visibility graphs, binary space partitioning and hidden surface removal (in theory and applications) |
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05.03.2008 Mateusz Juda |
Podstawy Informatyki Introduction to Constructive Logic and Mathematics (I) |
| Mateusz referuje pierwszą część skryptu Thomasa Streichera, pod tym samym tytułem, dostępnego TUTAJ. Możemy spodziewać się następujących zagadnień: - Natural Deduction for Constructive Logic - A Hilbert Style System for CL - Truth–Value Semantics of CL - Embedding Classical into CL (Constructive = Intuitionistic) |
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27.02.2008 Jan Jeżabek |
Informatyka Teoretyczna Online Buffer Management - Increasing Machine Speed |
| We consider the following problem: a network switch receives packets characterized by a deadline and a weight. In each step the switch can transmit a fixed number of packets (this number is called the speed of the machine). The goal of the machine is to maximize the sum of weights of the transmitted jobs. It is easily seen that any on-line algorithm is outperformed by an optimal off-line algorithm with the same speed on some instance. We will show that this is true even if the speed of the on-line algorithm is increased by an arbitrary factor with respect to the speed of the off-line algorithm. |
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27.02.2008 Paweł Waszkiewicz |
Podstawy Informatyki Konstruktywizm wg Bridgesa. Piękno wg Patarai. |
| Tematem naszego seminarium w tym semestrze jest konstruktywizm i intuicjonizm. Podczas pierwszego spotkania przedstawię artykuł Douglasa Bridgesa pt. Reality and Virtual Reality in Mathematics, w którym autor daje nam krótki kurs historii matematyki konstruktywnej. Aby zniszczyć humanistyczny nastrój, który nieuchronnie wytworzy się podczas wykładu historycznego, zakończę twardym, ale konstruktywnym dowodem faktu, że każda funkcja monotoniczna f:X->X na posecie zupełnym X, wyposażonym w element najmniejszy, posiada najmniejszy punkt stały. Dowód, pochodzący od gruzińskiego matematyka Dimitriego Patarai, nie używa liczb porządkowych. |
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23.01.2008 Piotr Zieliński |
Informatyka Teoretyczna Computing a longest common increasing subsequence |
| Computing a longest common increasing subsequence of two given sequence is classical problem in computer science. It can be solved in O(n*n*m) time and O(n*m) space using a simple dynamic programming technique. In 2004 I-Husan Yang proposed an O(n*m)-time algorithm based on the relationship between computing a longest common increasing subsequence and computing a longest common subsequence. Two years later, Yoshifumi Sakai improved the space complexity of Yang's algorithm and presented O(n+m)-space algorithm. I will talk about both algorithms, especially how to implement them efficiently. References:I-Hsuan Yang, A fast algorithm for computing a longest common increasing subsequence" , 2004 Yoshifumi Sakai, A linear space algorithm for computing a longest common increasing subsequence, 2006 |
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23.01.2008 Kacper Marcisz |
Podstawy Informatyki An application of stream calculus to signal flow graphs |
| Pan Kacper referujr artykuł Jana Ruttena pod tym samym tytułem. Dane bibliograficzne artykułu: Proceedings FMCO 2003 (Formal Methods for Components and Objects). Editors: F.S. de Boer, M.M. Bonsangue, S. Graf, W.P de Roever. Lecture Notes in Computer Science 3188, Springer Verlag, 2004, pp. 276-291. Jest to kontynuacja tematyki przedstawionej przez panią Dominikę Majsterek wcześniej na naszym seminarium. |
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16.01.2008 Jiří TůmaCharles University, Prague |
Informatyka Teoretyczna Recovering ENIGMA, the cipher system |
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16.01.2008 Szymon Wójcik |
Podstawy Informatyki Parallel reductions in lambda calculus |
| The notion of parallel reduction is extracted from the simple proof of the Church-Rosser theorem by Tait and Martin-Löf. Intuitively, this means to reduce a number of redexes (existing in a lambda term) simultaneously. During the talk, after reevaluating the significance of the notion of parallel reduction in Tait-and-Martin-Löf type proofs of the Church-Rosser theorems, we show that the notion of parallel reduction is also useful in giving short and direct proofs of some other fundamental theorems in reduction theory of lambda calculus. |
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14.01.2008 Karol Kosiński |
Algorytmiczne Aspekty Kombinatoryki On the Parity of Exponents in the Factorization of n! |
| It is shown that, for any k, there exist infinitely many positive integers n such that in the prime power factorization of n!, all first k primes appear to even exponents. This answers a question of Erdos and Graham ("Old and New Problems and Results in Combinatorial Number Theory", L'Enseignement Mathematique Imprimerie Kundia, Geneva, 1980). A few generalizations are provided as well. |
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09.01.2008 Marek ChrobakUniversity of California at Riverside |
Informatyka Teoretyczna Doubling technique in approximation algorithms |
| This talk is based on an expository article by Claire Kenyon-Mathieu and myself, in which we show that there is a number of approximation algorithms in the literature that use essentially the same (but yet not explicitely identified) technique that we refer to as "doubling". The essence of this approach is to use exponentially increasing estimates on the optimal solution to design approximate solutions. It can be used both in offline and online approximation algorithms. Applications include several clustering, searching, facility location, and scheduling problems. |
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09.01.2008 Dominika Majsterek UJ |
Podstawy Informatyki Behavioural differential equations: a coinductive calculus (część 2) |
| Pani Dominika referuje artykuł: J.J.M.M. Rutten Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theoretical Computer Science vol. 308(1-3), pp. 1-53, 2003. |
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19.12.2007 Mikołaj Bojańczyk, Instytut Informatyki, Uniwersytet Warszawski |
Podstawy Informatyki Automaty ścieżkowe |
| Automat ścieżkowy to rodzaj automatu skończonego na drzewach. W odróżnieniu od powszechnie używanych automatów na drzewach, automat taki przetwarza drzewo sekwencyjnie, a nie równolegle. Pokażę, że automat ścieżkowy mało potrafi. |
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17.12.2007 Katarzyna Grygiel |
Algorytmiczne Aspekty Kombinatoryki On the chromatic number and independence number of hypergraph products |
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12.12.2007 Dominika Majsterek |
Podstawy Informatyki Behavioural differential equations: a coinductive calculus |
| Pani Dominika referuje artykuł: J.J.M.M. Rutten Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theoretical Computer Science vol. 308(1-3), pp. 1-53, 2003. Abstract: We present a theory of streams (infinite sequences), automata and languages, and formal power series, in terms of the notions of homomorphism and bisimulation, which are cornerstones of the theory of (universal) coalgebra. This coalgebraic perspective leads to a unified theory, in which the observation that each of the aforementioned sets carries a so-called final automaton structure, plays a central role. Finality forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductiove definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the aforementioned stuctures, closely resembling calculus from classical analysis. |
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10.12.2007 Leszek Horwath |
Algorytmiczne Aspekty Kombinatoryki Multicolored forests in bipartite decompositions of graphs |
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05.12.2007 Libor BartoEduard Čech Center, Prague |
Informatyka Teoretyczna The algebraic approach to Constraint Satisfaction Problem, recent progress |
| Many decision problems in combinatorics, computer science, artificial intelligence, logic, etc. can be expressed as so called Constraint Satisfaction Problems (CSPs). For a fixed relational structure R, CSP(R) is the following decision problem: INPUT: A relational structure $S$ of the same signature as R OUTPUT: Is there a homomorphism ftom S into R The central problem in this area is the Dichotomy Conjecture of Feder and Vardi stating that, for any relational structure R, CSP(R) is either solvable in polynomial time or NP-complete. I will talk about the universal algebraic approach to this problem and mention some recent developments.
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05.12.2007 Tytus Bierwiaczonek |
Podstawy Informatyki Logical aspects of finite automata |
| Pan Tytus referuje przeglądowy artykuł Wolfganga Thomasa pt. Languages, Automata and Logic. Usłyszymy o zależnościach pomiędzy automatami i monadyczną logiką drugiego rzędu. Artykul dostępny jako [Tho96] na stronie http://www.automata.rwth-aachen.de/publications/pub-Thomas.html albo od razu tutaj . |
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03.12.2007 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki Finding disjoint paths in expanders deterministically and online |
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28.11.2007 21.11.2007 14.11.2007 Andrzej Pezarski |
Informatyka Teoretyczna On-line clique covering of proper interval graphs |
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28.11.2007 Thierry Joly |
Podstawy Informatyki Undeciding lambda-definability again |
| The Definability Problem (DP for short) is the question whether a given functional of some hereditarily finite type structure over a single atomic type is the interpretation of a closed lambda-term or not. DP was first considered about Full Type Structures by G. Plotkin in 1973, [Plo73]. R.Statman [Sta82] pointed out that deciding it would solve at once quite a few existential problems of the typed lambda-calculus, the most famous of which is the (still open) Matching Problem: "Given lambda-terms A, B, is there a lambda-term X such that AX=B?" Then DP became a little Graal, finally proved undecidable by R. Loader in 1993, [Loa01]. Is that the end of the story? One may object that in smaller models than the Full Type Structures, the few lambda-definable functionals would not be so easily lost and that deciding definability in any class of (smaller) models that is strongly complete with respect to the lambda-calculus (in the sense of [Sta82]) would also yield the benefits pointed out by Statman. Unfortunately, we will show in this talk that the restriction of DP to a fixed model M is actually undecidable for a fair amount of models M, including all the non trivial stable models and order extensional models, except possibly the 2 element Scott model. These stronger results were obtained by cleaning previous proofs and by identifying their efficient ingredients. This work of simplification also yields a particularly short and easy proof of the undecidability of DP for Church pure typed lambda-calculus that will first be detailed. [Plo73] Gordon Plotkin. Lambda-definability and logical relations. Memorandum SAI-RM-4, University of Edinburgh, 1973. [Sta82] Richard Statman. Completeness, invariance and lambda-definability. JSL 47:17-26, 1982. [Loa01] Ralph Loader. The Undecidability of lambda-Definability. In Logic, Meaning and Computation: Essays in Memory of A. Church, 331-342, Anderson & Zeleny editors, Kluwer Acad. Publishers, 2001. |
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26.11.2007 Mikołaj Pudo |
Algorytmiczne Aspekty Kombinatoryki Upper and lower bounds for satisfiability threshold |
| The 3-SAT problem consists in determining if a boolean formula with 3 literals per clause is satisfiable. When the ratio between the number of clauses and the number of variables increases, a threshold phenomenon is observed: the probability of satisfiability appears in simulations to decrease sharply from 1 to 0 in the neighbourghood of a threshold value, conjectured to be close to 4.25. In my talk I will present approach to upper and lower bounds for the threshold's potential location based on urn models, and generating functions. |
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21.11.2007 Tomasz Połacik, Instytut Matematyki, Uniwersytet Śląski, Katowice |
Podstawy Informatyki Problemy modeli Kripkego dla teorii pierwszego rzędu |
| Jest faktem ogólnie znanym, że modele Kripkego stanowią ważne i efektywne narzędzie służące do badania intuicjonistycznych teorii pierwszego rzędu. Na przykład, znane są ich liczne interesujące zastosowania w przypadku takich konstruktywnych teorii jak arytmetyka Heytinga czy intuicjonistycza teorii mnogości Kripkego-Platka. Na uwagę zasługuje jednak fakt, że w przeciwieństwie do sytuacji teorii modeli klasycznych, wciąż brakuje ogólnych metod i konstrukcji dla modeli Kripkego. Przypomnijmy, że - nieformalnie - na model Kripkego możemy patrzeć jak na rodzinę klasycznych modeli dla danego języka pierwszego rzędu, w której określony jest porządek wyznaczony przez homomorfizmy modeli tej rodziny. Na całej tej strukturze zdefiniowane jest pojęcie spełnialności. Przy czym, w odróżnieniu od klasycznej spełnialności (w sensie Tarskiego) traktowanej lokalnie, w pojedynczym modelu rozważanej rodziny, spełnialność zdefiniowana na modelu Kripkego jest spełnialnością intuicjonistyczną. Nietrudno jest zauważyć, że model Kripkego wyznaczony przez pojedynczy klasyczny model M z identycznościowym homomorfizmem może być utożsamiony z M widzianym jako model klasyczny. W tym sensie, pojęcie modelu Kripkego można uważać za uogólnienie klasycznego pojęcia modelu pierwszego rzędu. W sposób naturalny powstaje więc kwestia stosownego uogólnienia pojęć i związków klasycznej teorii modeli na przypadek modeli Kripkego. Jedno z podstawowych zagadnień teorii modeli dotyczy elementarnej równoważności. W referacie rozważony zostanie problem elementarnej równoważności w odniesieniu do modeli Kripkego, a w celu jego rozwiązania, wprowadzone zostanie pojęcie bisymulacji dla modeli Kripkego pierwszego rzędu. Pojęcie to jest, z jednej strony, rozszerzenieniem pojęcia bisymulacji dla modeli Kripkego dla intuicjonistycznej logiki zdań oraz, z drugiej strony, uogólnieniem - w opisanym wyżej sensie - pojęcia gry Ehrenfeuchta dla klasycznych modeli pierwszego rzędu. Oprócz omówienia podstawowych własności bisymulacji, zostaną zaprezentowane również jej zastosowania. W szczególności, przedstawiona zostanie konstrukcja elementarnego podmodelu modelu Kripkego. |
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19.11.2007 Filip Sokołowski |
Algorytmiczne Aspekty Kombinatoryki Unprovability of the 3n+1-conjecture |
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14.11.2007 07.11.2007,Lech Duraj, Grzegorz Gutowski |
Informatyka Teoretyczna Optimal orientation on-line |
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14.11.2007 Mateusz Kostanek |
Podstawy Informatyki Tree walking automata cannot be determinized |
| Mateusz zreferuje artykul Mikołaja Bojańczyka i Thomasa Colcombeta, który otrzymał tytuł ,,best paper'' na konferencji ICALP 2004. Abstrakt: Tree-walking automata are a natural sequential model for recongnizing languages of finite trees. Such automata walk around the tree and may decide in the end to accept it. It is shown that deterministic tree-walking automata are weaker than nondeterministic tree-walking automata. |
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07.11.2007 24.10.2007,Mariusz Łusiak |
Podstawy Informatyki On learning monotone Boolean functions under the uniform distribution |
| We prove two general theorems on monotone Boolean functions which are useful for constructing a learning algorithm for monotone Boolean functions under the uniform distribution. The first result is that a single variable function f(x) = x_i has the minimum correlation with majority function among all fair monotone functions. The second result is on the relationship between the influences and the average sensitivity of a monotone Boolean function. The talk is based on a paper by Kazuyuki Amano and Akira Maruoka. |
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29.10.2007 Michał Zmarz |
Algorytmiczne Aspekty Kombinatoryki Complexity of nonrepetitive colorings |
| A coloring of a graph is "nonrepetitive" if no path contains two identical adjacent blocks. A resent result by Marx and Schaefer asserts that testing whether a given coloring is nonrepetitive is coNP-hard. However, if one restricts to blocks of length at most k then the problem becomes fixed-parameter tractable. |
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24.10.2007 17.10.2007 10.10.2007 Bartosz Walczak |
Informatyka Teoretyczna Algorithmic meta-theorems and treewidth |
Algorithmic meta-theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. We focuse on the model checking problem, which is to decide whether a given graph satisfies a given formula. Some restrictions of this problem to specific classes of graphs (with bounded treewidth, excluded minors etc.) turn out to be fixed-parameter tractable (fpt). We define combinatorial notions of tree decomposition, treewidth and local treewidth, and prove that MSO model checking on graphs with bounded treewidth and FO model checking on graphs with bounded local treewidth are fpt. The latter result uses Gaifman's Locality Theorem, which in one of basic tools in finite model theory. The talks are based on a paper by Martin Grohe [4]. |
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22.10.2007 Wiktor Żelazny |
Algorytmiczne Aspekty Kombinatoryki On graphs represented by colored intervals |
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17.10.2007 Mateusz Juda |
Podstawy Informatyki Automata for XML - A survey |
| Mateusz referuje artykul Thomasa Schwenticka pod tym samym tytułem z Journal of Computer and System Sciences 73(2007), str. 289-315. Abstract artykułu: Automata play an important role for the theoretical foundations of XML data management, but also in tools for various XML processing tasks. This survey article aims to give an overview of fundamental properties of the different kinds of automata used in this area and to relate them to the four key aspects of XML processing: schemas, navigation, querying and transformation. |
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15.10.2007 Maciej Chociej |
Algorytmiczne Aspekty Kombinatoryki A randomised scheme for geometrical algorithms |
| An abstract, randomised scheme for structure creating algorithms can be used in solving many geometrical problems. One of those is obtaining a Delaunay triangulation of a given set of points. For a set P of points in a d-dimensional Euclidean space a Delaunay triangulation is a triangulation T(P) such that no point in P is inside the circum-hypersphere of any simplex in T(P). The general scheme and an exemplary Delaunay triangulation algorithm shall be presented with some foreword on other applications of the scheme and derandomization of resulting algorithms. |
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08.10.2007 Jarek Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Invisible runners in finite fields |
| I will present some results on the Lonely Runner Problem in a setting of finite fields, discuss connections to graph coloring, matroid flows, and view obstruction, and offer several new open problems. |
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03.10.2007 Jan Jeżabek |
Informatyka Teoretyczna ICALP, LICS, LCC 2007 |
| Selected results and open problems from ICALP, LICS, LCC 2007 are presented |
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03.10.2007 Jakub Kozik |
Podstawy Informatyki Intuitionistic versus Classical Tautologies |
| We consider propositional formulas built on implication. The size of a formula is measured by the number of occurrences of variables. We assume that two formulas which differ only in the naming of variables are identical. For every n we investigate the proportion between the number of intuitionistic tautologies of size n compared with the number of classical tautologies of that size. We prove that the limit of that fraction is 1 when n tends to infinity. |
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13.06.2007 Mateusz Kostanek |
Informatyka Teoretyczna On k-server problem |
| The on-line k-server problem is set on a metric space inhabited by k servers. Initially, each sever is positioned at some point of the space. Over time, request arrive for service at points of the space. Immediately after a request at some point q comes in, a server must be moved to q in order to serve the request. When a server moves it incurs a cost equal to the distance it covers. Our goal is to design on-line algorithm which will decide which server to move when a request arrives so that any sequence of request can be served with cost as small as possible. In this talk we will present the work function algorithm for the k-server problem and prove that it has competitive ratio at most 2k-1 References:M. S. Manase, L. A. McGeoch, And D. D. Sleator: Competitive algorithms for on-line problems E. Koutsoupias, C. Papadimitriou: On th k-Server conjecture |
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06.06.2007 Josh Buresh-OppenheimSimon Fraser University |
Informatyka Teoretyczna Formalizing Algorithmic Paradigms |
| Since most algorithms that people use can intuitively be classified into large paradigms of algorithms such as greedy, dynamic programming, linear and semidefinite programming, local search, etc., it is natural to ask what problems can be computed by these paradigms. This question can be seen as an alternative to asking what problems can be computed by all, say, poly-time algorithms in that the restriction on the algorithms is more conceptual rather than complexity-theoretic. As we will illustrate, it is also a question of vital importance to algorithm designers. Of course, to ask a question about an algorithmic paradigm, you first have to formally define the paradigm. We offer one very natural model, pBT, which captures many algorithms generally considered to be dynamic programming or backtracking. We demonstrate upper and lower bounds in this model for problems such as interval scheduling and SAT. We also present a very powerful model for linear and semidefinite programming due to Lovasz and Schrijver and show some strong lower bounds for SAT. |
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16.05.2007 Edyta SzymańskaUAM |
Informatyka Teoretyczna The complexity of perfect matchings in hypergraphs |
| Given a k-uniform hypergraph H=(V,E) on n vertices, we define a perfect matching as a set of $\lfloor n/k\rfloor$ disjoint edges in E(H). From the algorithmic perspective, a few natural problems regarding this notion can be considered. One is a decision problem asking whether a given k-uniform hypergraph contains a perfect matching, which is NP-complete for k>2. In view of this fact, a question arises, under which additional conditions for a k-uniform hypergraph there exists a polynomial time algorithm finding a perfect matching in it. We define the minimum collective degree $\delta_{k-1}(H)$ to be the largest integer d such that every (k-1)-element set of vertices of H belongs to at least d edges of H. In this talk we will present an algorithm which finds a perfect matching in a k-uniform hypergraph of the minimum collective degree roughly n/2 in polynomial time. On the negative side, we will prove that the problem of deciding whether a given k-uniform hypergraph H with $\delta_{k-1}(H)> c|V(H)|$ for c<1/k contains a perfect matching is NP-complete. |
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15.05.2007 Andrzej Ruciński (UAM) |
Algorytmiczne Aspekty Kombinatoryki Perfect matchings in uniform hypergraphs |
| I will present a recent result, jointly with Rodl and Szemeredi, establsihing an exact degree threshold for the existence of a perfect matching in a $k$-uniform hypergraph. Some algorithmic aspects of this problem will be discussed in the lecture by Edyta Szymanska, next day on TCS seminar. |
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25.04.2007 18.04.2007 11.04.2007 Mateusz Juda |
Informatyka Teoretyczna Hypergraphs isomorphism problem |
Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn't be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this talk, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions |
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28.03.2007 21.03.2007 14.03.2007 07.03.2007 Kamil Kloch Piotr Micek Grzegorz Matecki |
Informatyka Teoretyczna On-line chain partitioning of semi-orders |
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27.03.2007 Lech Duraj |
Algorytmiczne Aspekty Kombinatoryki Elusive graph properties |
| Let P be a fixed property of graphs (planarity, 3-colorability, connectedness, etc.). The following game is played by two players (Adam and Eve) on the vertex set V = {1,2,...,n}. In one round Adam asks a question of the form: "is there an edge between i and j?", and Eve answers: "yes" or "no". Adam's goal is to learn as quickly as possible whether the constructed graph will have property P, or not. Eve wants to keep Adam in uncertainty until the very last question. In this case she is a winner and property P is called "elusive". The main conjecture states that every non-trivial monotone graph property is elusive. The most impressive result so far confirms the conjecture when n is a prime power. The proof uses topological arguments. References:A. Björner, Topological methods. Handbook of combinatorics, Vol. 1, 2, 1819--1872, Elsevier, Amsterdam, 1995. |
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27.03.2007 Jarek Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Diophantine approximation, graph coloring, and the lonely runner problem |
| Suppose n runners are running with constant speeds around a circle of circumference 1. A runner is "lonely" at moment t if there are no other runners within a circular distance 1/n from his/her actual position. Is it true that for every set of n different speeds a lonely runner always appears? This innocently looking question is open for more than six runners and has some intriguing connections to diophantine approximation and graph coloring. |
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06.03.2007 Paweł Walter |
Algorytmiczne Aspekty Kombinatoryki Splitting Necklaces |
| Two thieves stolen a necklace with even number of beads in each of r colors. They want to split it so that each of them could get the same number of beads in each color. How many cuts are needed in the worst case? Clearly, if the necklace is open and beads in one color form a segment then r cuts are necessary. Curiously, this number is always sufficient, as can be proved using the Borsuk-Ulam theorem. The problem has continuous and multiple versions for which topological method is also applied. |
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28.02.2007 Maciej Gazda |
Informatyka Teoretyczna Polar SAT and related graphs |
| References:Igor Zverovich, Olga Zverovich: "Polar SAT and related graphs" |
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18.01.2007 Grzegorz Gutowski |
Algorytmiczne Aspekty Kombinatoryki Kneser's Conjecture |
| Let G(n,k) denotes a graph whose vertices are k-element subsets of {1,2,...,n}, with two vertices adjacent iff they are disjoint. It is easy to see that G(n,k) is (n-2k+2)-colorable, and Kneser conjectured that actually that many colors are needed. The conjecture was proved by Lovasz by using topological methods. A simple proof based on the Borsuk-Ulam theorem, found later by Barany, will be presented. |
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17.01.2007 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Combinatorial applications of the Borsuk-Ulam theorem |
| A set S of 3n points in general position (no four on a plane) is given in 3-dimensional space. Suppose the points are colored red, blue, and green so that there are exactly n points in each color. Can we partition the set S into n triangles so that 1) each triangle is multicolored (i.e. no two of its vertices have the same color), 2) no two triangles (considered as convex hulls of their vertices) intersect? The answer is yes, but the only known proof of this fact uses topological argument - the famous Borsuk-Ulam theorem, known also as the Ham Sandwich Theorem. We will present more of such unexpected applications of topology in combinatorics. References:A. Björner, Topological methods. Handbook of combinatorics, Vol. 1, 2, 1819--1872, Elsevier, Amsterdam, 1995. Alon, Noga Nonconstructive proofs in combinatorics. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1421--1429, Math. Soc. Japan, Tokyo, 1991. |
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17.01.2007 Tomasz Jurkiewicz |
Informatyka Teoretyczna Towards a trichotomy for quantified H-coloring |
| A very natural generalisation of graph coloring problems is defined in terms of graph homomorphism; the problem takes as input a graph G and accepts it if, and only if, there exists a homomorphism into a fixed graph H. This problem is known as the H-coloring problem and is tractable if H is bipartite and NP-complete otherwise. Quantified H-coloring problem is definable via two-player games and is tractable if H is bipartite; NP-complete if H is not bipartite and not connected; and, PSPACE-complete if H is connected and contains a unique cycle which is of odd length. (Conjecture: Problem is PSPACE-complete if H is not bipartite and connected.) References:B.Martin and F.Madeleine, Towards a trichotomy for quantified H-coloring |
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03.01.2007 13.12.2006,Wiktor Żelazny |
Informatyka Teoretyczna Recognizning Interval Bigraphs |
| We introduce the class of interval bigraphs, bipartite graphs similiar to interval graphs. We present algorithm that recognizes these graphs in polynomial time, shown by Muller in 1993. Also a characterization of interval bigraphs in terms of their complement graphs due to Hell and Huang (2003) will be presented. |
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20.12.2006 Jarek Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Games and sequences |
| This will be a survey talk presenting a collection of open problems on combinatorial games and integer sequences. |
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14.12.2006 Lech Duraj |
Algorytmiczne Aspekty Kombinatoryki Firing chips on graphs |
| Suppose each vertex v of a graph G is assigned with some number of chips c(v). If c(v) is at least the degree of v then v can be "fired", and in effect each neighbor of v will receive one chip from v. In this way initial configuration of chips evolves and may eventually reach a stable form in which no vertex can be fired. Many natural question can be asked: which configurations are eventually stable? How long it may take to reach a stable configuration? What properties has a (naturally defined) poset of configurations? |
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13.12.2006 Przemysław Broniek |
Algorytmiczne Aspekty Kombinatoryki Ranking tournaments |
| For a given digraph D, let f(D) be the minimum number of edges whose reversal or removal turns D into an acyclic digraph. It was known for over thirty years that the problem of determining f(D) is NP-hard. Recently Noga Alon proved that it is NP-hard even for tournaments (that is complete oriented graphs). The proof makes use of the previous fact and pseudorandom properties of Paley tournaments. References:N. Alon, Ranking tournaments, SIAM J. Discrete Math. 20 (2006), 137-142. |
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30.11.2006 Andrzej Pezarski |
Algorytmiczne Aspekty Kombinatoryki Chip firing games |
| A pile of n chips occupies a vertex v of a long path. In one move of the game we split the pile into two equal parts and place them into neighbors of v (leaving one chip at v if n was odd). We repeat this move, each time choosing a vertex to be "fired" arbitrarily. The game ends if there is at most one chip on every vertex. Is it true that we always end with a stable configuration of chips? There are many variants of the game leading to iteresting and deep mathematical concepts. References:A. Björner, L. Lovász, P.W. Shor, Chip-firing game on graphs, European J. Combin. 12 (1991) 283--291. A. Björner, L. Lovász, Chip-firing games on directed graphs, J. Algebraic Combin. 1 (1992), no. 4, 305--328. G. Tardos, Polynomial bound for a chip firing game on graphs, SIAM J. Discrete Math. 1 (1988), no. 3, 397--398. R. Anderson, L. Lovász, P. Shor, J. Spencer, E. Tardos, and S. Winograd, Disks, balls, and walls: Analysis of a combinatorial game, Amer. Math. Monthly 96 (1989), pp. 481-- 493. |
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29.11.2006 Piotr Micek |
Algorytmiczne Aspekty Kombinatoryki Dice, boxes, and majority tournaments |
| Suppose we have 2k-1 linear orders of a finite set V. A k-majority tournament T on V is defined so that u dominates v in T iff u lies above v in more than a half of the orders. Let F(k) be the maximum over all k-majority tournaments T of the size of a minimum dominating sets of T. Using geometric arguments, it can be proved that F(k) is finite for every k. For instance, it is not hard to show that F(2)=3, but this is the last exact value of F(k) determined so far. References:N. Alon, G. Brightwell, H. A. Kierstead, A. V. Kostochka, P. Winkler, Dominating sets in k-majority tournaments, J. Combin. Theory (ser. B), 96 (2006), 374-387. |
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29.11.2006 22.11.2006,Jacek Krzaczkowski |
Informatyka Teoretyczna Complexity of term equation problem, cntd. |
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23.11.2006 Piotr Micek |
Algorytmiczne Aspekty Kombinatoryki Trees, tournaments, and the Second Neighborhood Conjecture |
| The second neighborhood conjecture states that every directed graph has a vertex v such that the number of vertices that can be reached from v by exactly two jumps (but not in one jump) along directed edges is at least as the number of its out-neighbors. Most probably this is true, but nobody could prove it, as yet. A simple proof will be presented that the conjecture holds for tournaments. |
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22.11.2006 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki The permanent lemma |
| Let A be a square matrix of size n. The permanent lemma asserts that if per(A) is non-zero, then there is a vector X, whose components can be chosen from any prescribed sets of size 2, such that the vector AX is nowhere zero. This looks somewhat technical, but there are many combinatorial problems that can be expressed in this way. For instance, one can show that nonvanishing of permanents of certain matrices is equivalent to the Four Color Theorem. |
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09.11.2006 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki 123-conjecture |
| Let G be a connected graph with at least three vertices. Suppose we assign one of the integers 1, 2, or 3 to each edge of G. In this way each vertex v in G obtains the sum of numbers lying on the edges incident to v. Such an assignment is proper if no two adjacent vertices have the same sum. Can we always do a proper assignment using just three numbers 1, 2, and 3? It is known that this can be done using 1, 2, …, 16, and that for almost every graph G, only 1 and 2 are sufficient. There are many related open questions. |
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08.11.2006 Jarosław Grytczuk |
Algorytmiczne Aspekty Kombinatoryki Nonrepetitive colorings of graphs |
| A coloring of the vertices of a graph G is nonrepetitive if one cannot find a color pattern of the form AA on any simple path of G, where A is any sequence of colors. The minimum number of colors needed is the Thue chromatic number of G, denoted by T(G). It is known that T(G) is bounded for graphs of bounded maximum degree, bounded treewidth, and for graphs with forbidden planar minor. The major open problem is whether there exists a finite constant N (no matter how huge) such that T(G) is at most N for every planar graph G. There are lots of unexpected connections to other chromatic parameters, as well as plenty of related unsolved questions. (Joint work with Noga Alon) |
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08.11.2006 Stefan Felsner TU Berlin |
Informatyka Teoretyczna Embeddings of Planar Graphs |
| A graph is planar if it admits a crossing-free drawing in the plane. In the first instance, such a drawing can be everything but nice. I sketch approaches to obtain nice drawings. A particularly elegant method goes back to work of W. Schnyder. He succeded in producing straight-line embeddings of planar graphs on small grids. I show how Schnyder's ideas continue to produce new insights and results. In particular it turns out that good drawings in the plane can be obtained via a detour through dimension three. |
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25.10.2006 Tomasz Gorazd |
Informatyka Teoretyczna Complexity of term equation problem |
| We study the computational complexity of the problem of satisfiability of equation between terms over a finite algebra (TERM-SAT). We describe many classes of algebras where the complexity of TERM-SAT is determined by the clone of term operations. We classify the complexity for algebras generating the maximal clones. Using this classification we describe a lot of algebras where TERM-SAT is NP-complete. We classify the situation for clones generated by an order or a permutation relation. We introduce the concept of semiaffine algebras and show polynomial time algorithms solving the satisfiability problem for them. |
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20.10.2006 Hal KiersteadArizona State University |
Informatyka Teoretyczna 18:15 On-line Ramsey Theory |
| Two players, Builder and Painter, play the following game on a fixed set of vertices V. In one round Builder presents an edge e linking two previously independent vertices of V and Painter paints e using one of c colors. Builder's goal is to force Painter to create a monochromatic copy of a fixed graph F. If there are no other restrictions then Painter has no chances to avoid F (by Ramsey's theorem). But what if Builder is not allowed to construct a graph whose chromatic number exceeds that of F? We prove that even with this obstacle Builder wins this game for any number of colors. Our main tool is an auxiliary "Ramsey survival game", which is interesting in it's own right. (Joint work with Goran Konjevod) |
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11.10.2006 Arkadiusz Pawlik |
Informatyka Teoretyczna Integer programming and counting lattice points in rational convex polyhedra |
| It is possible to solve the linear programming problem in polynomial time, but if we require that the solution is integral, then the problem becomes NP-hard. However, as shown by Lenstra in 1983, if we fix the number of variables, then the problem is in P. I present a more recent approach which involves counting the solutions with generating functions. References: Alexander Barvinok, James E. Pommersheim, An Algorithmic Theory of Lattice Points in Polyhedra |
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28.06.2006 21.06.2006 14.06.2006 Grzegorz Matecki |
Informatyka Teoretyczna On-line graph coloring on a bounded board |
We consider a version of on-line graph coloring problem as a two person game with some additional conditions for players. Players are called Spoiler and Painter. Spoiler reveals a graph by putting or removing a node. But at each time the total number of nodes is bounded. Painter must assign a color to any new node such as two nodes connected by an edge have different colors. He cannot change colors of already colored nodes. |
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07.06.2006 31.05.2006,Bartosz Walczak |
Informatyka Teoretyczna Flows in skew-symmetric networks |
| The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. The idea behind the solution is to find a good initial flow and then to augment the flow along so-called regular paths. The initial flow can be a slightly modified antisymmetrization of an ordinary maximum flow. Finding regular augmenting paths is based on Edmonds' "blossom" algorithm for finding a maximum matching. The resulting algorithm for MSFP is competitive with the fastest known algorithms for the maximum flow and maximum matching problems. References:A. V. Goldberg, A. V. Karzanov. "Maximum skew-symmetric flows and matchings". Mathematical Programming 100, 537-568 (2004) http://www.avglab.com/andrew/pub/skew-max.pdf A. V. Goldberg, A. V. Karzanov. "Path problems in skew-symmetric graphs". Combinatorica 16, 127-174 (1996) http://ftp.cs.stanford.edu/cs/theory/goldberg/skew-sp.ps.Z (preliminary version) |
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24.05.2006 Jarosław Grytczuk,Zielona Góra |
Informatyka Teoretyczna Graph coloring games for daltonists |
| Ann and Ben are coloring alternately the vertices of a graph G using a fixed set of colors, with Ann playing first. They both have to respect the rule of a proper coloring, that is, none of them is allowed to create a monochromatic edge at any moment of the game. Ann's goal is to color the whole graph successfully, in which case she is a winner. Ben's goal is of course different: he perfidiously tends to create a partial coloring that cannot be extended to the whole graph, without introducing new colors. In this case he is a winner. The game chromatic number of a graph G, denoted g(G), is the least number of colors guaranteeing a win for Ann. The main open problem is to find out how large is g(G) for planar graphs. Curiously, currently best strategy allows Ann to win (with 17 colors) even as a completely color blind person! I will present this and other techniques in a most recent treatise. Joint work with Tomasz Bartnicki, Hal Kierstead and Xuding Zhu |
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10.05.2006 Tadeusz Prochwicz |
Informatyka Teoretyczna Algorithms for four variants of the exact satisfiability problem |
| References:V.Dahllof, P.Jonsson and R.Biegel, Algorithms for four variants of the exact satisfiability problem, Theoretical Computer Science, 320(2004), 373-394. |
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26.04.2006 Gabor Kun,Loránd Eötvös University, Budapest |
Informatyka Teoretyczna Forbidden patterns and homomorphism problems |
| We say that two subclasses of NP are (computationally) equivalent if for every language in one class there is a polynomially equivalent one in the other class. A typical equivalence class is the class of k-colouring problems (k in N), it is always in P or NP-complete. NP turned out not to be equivalent to this class (unless P=NP): there is a problem in NP which is neither in P nor NP-complete. We show some types of combinatorial problems like edge colourings or graph decompositions expressing the full computational power of the NP class. Hence these problem classes contain also some problems of "exotic" complexity. The first natural candidate that seems not to be equivalent to NP is the class called MMSNP (Monotone Monadic Strict NP) or Forbidden patterns problem. (A typical example for a language in the class: graphs with vertex set partitionable into two subsets without triangle.) This class is conjectured to contain only NP-complete and polynomial time solvable problems. We prove that the class MMSNP can be expressed in the simpler terminology of relational structure homomorphism problems (called Constraint Satisfaction Problems): such a language contains for a fixed structure A the relational structures that can be mapped to A. homomorphically. The first such result was only a random equivalence. The proof of the deterministic equivalence uses some type of expander structures, a typical tool in derandomization. |
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19.04.2006 Jarosław Duda |
Informatyka Teoretyczna Optimal coding by random algorithms |
| Each point of the plane grid Z^2 is labelled by 1 bit : "0" or "1". Forbiding two 1's to be adjacent reduces average information capacity (i.e., entropy) to 0.588 bit/point. The talk gives an intuitive background to symmetry, theory of information and statistical approach to combinatorial problems. We address and discuss the following problems: - how typical labeling looks like? - how to generate these labelings? - how to use these labelings to encode information? - how to use this stuff in practice? |
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29.03.2006 22.03.2006,Andrzej Soroczyński |
Informatyka Teoretyczna Ulam games |
| Our investigation deal with searching lair game. Game was proposed by Ulam, and that is why we call it Ulam searching game with fixed number of lies. In this game one person (we call her Carole) thinks about one number from 1 to n. Second player (we call him Paul) is asking about some subset of {1,2,...,n}, and Carole reply if number she is thinking about is a member of Paul's subset. Carol is allowed to lie l(fixed number) times. Let L(l,M) be the minimum number of questions which Paul must ask to win. The main results of presented papers are: 1. For each l there exist such M that for all N >= M the following is true: L(l,2N) <= L(l,N) + 2. 2. For each l there exist such M that for all N >= M the following is true: L(l,3/2*N) <= L(l,N) + 1. References:C. Deppe, Strategies for the Renyi-Ulam Game with fixed number of lies, Theoret. Comput. Sci. 314 (2004), 45-55 J. Spencer, Ulam's searching game with fixed number of lies, Theoret. Comput. Sci. 95 (1992), 307-321 |
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22.02.2006 Andrzej Pezarski |
Informatyka Teoretyczna On line clique covering of proper interval graphs presented in a connected way |
| Proper interval graphs are graphs for which there is a representation by intervals of real line in which no interval is contained in another. After an easy observation that all greedy algorithms have competive ratio bigger than 2 we consider all possible algorithms. Now the situation is harder: a proof that 8/5 is a lower bound in this situation will be presented. |
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25.01.2006 18.01.2006 11.01.2006 Marcin Kozik |
Informatyka Teoretyczna 2EXPTIME-complete membership problems in algebra |
We construct a finite algebra generating a variety with PSPACE-complete membership problem first. Then we show another algebra with exponentially growing gamma function. In the final construction we use both of the previously mentioned algebras to produce a finite algebra that is able to model a computation of a Turing machine on an exponentially long tape. This gives an example of a finite algebra with EXPSAPCE-hard membership problem (on the other hand this problem is known to be in a 2-EXPTIME class). |
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04.01.2006 Wojciech Jawor,University of California, Riverside |
Informatyka Teoretyczna Job Scheduling in Next Generation Computer Networks |
| Two online job scheduling problems arising in next generation computer networks are discussed. In the first problem [3] the goal is to schedule n jobs on m identical machines, without preemption, so that the jobs complete in the order of release times and the maximum flow time is minimized. This problem arises in network systems with aggregated links, when it is required that packets complete their arrivals at the destination in the order of their arrivals at the receiver. This requirement is imposed by the IEEE 802.3 standard describing link aggregation in Local Area Networks. We present a deterministic algorithm Block with competitive ratio O(\sqrt{n/m}) and show a matching lower bound even for randomized algorithms. The second problem [1,2] is an online unit-job scheduling problem arising in networks supporting Quality of Service. Jobs are specified by release times, deadlines and nonnegative weights. The goal is to maximize the total weight of jobs, that are scheduled by their deadlines. We show that there does not exist a deterministic algorithm with competitive ratio better than 1.618 (the golden ratio). We also give a randomized algorithm with competitive ratio 1.582, showing that randomized algorithms are provably better than deterministic algorithms for this problem. References:F.Y.L.Chin, M.Chrobak, S.P.Y.Fung, W.Jawor, J.Sgall and T.Tichy, Online Competitive Algorithms for Maximizing Weighted Throughput of Unit-JobsW.Jawor, M.Chrobak and Ch.Dürr, Competitive Analysis of Scheduling Algorithms for Aggregated Links |
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14.12.2005 Maciej Żenczykowski |
Informatyka Teoretyczna Online interval coloring and variants |
| References:L. Epstein, M. Levy Online interval coloring and variants, Proc. of the 32nd ICALP (2005), 602-613 |
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07.12.2005 |
Informatyka Teoretyczna Problems from Programming Competitions in Poznan 2005 |
| References:Contest home page: http://www.mwpz.poznan.pl/resources.php |
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30.11.2005 Piotrek Micek |
Informatyka Teoretyczna The lower bound for on-line cliques covering for K_s-free graphs |
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23.11.2005 16.11.2005 19.10.2005 Iwona Cieslik |
Informatyka Teoretyczna On-line coloring for graphs with forbidden subgraphs |
It is known that the on-line coloring problem for arbitrary graphs is not competitive. However, restricting to special families of graphs, that have forbidden induced subgraphs (of some kind), the spoiler has his hands tied and the number of colors used by some on-line algorithms can be substantially reduced. We call these subgrahs: forcing structures. In my work I try to make a classification of competitive functions for various kind of families of graphs and also appoint the forcing structures for the on-line graph coloring. I was mostly looking for competitiveness for the graphs in the form of H-free: K_s-free, K_s,t-free, C_4-free, P_5-free and also for perfect and k-chordal graphs. |
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12.10.2005 Kamil Kloch |
Informatyka Teoretyczna EuroComb 2005 - Open Problems |
| A few open problems from the recent EuroComb conference (http://www.math.tu-berlin.de/EuroComb05/) is presented. These include the L(p, 1) labelling of graphs and the excluded subposets in the Boolean lattice. |
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05.10.2005 Lech Duraj |
Informatyka Teoretyczna Interval orders and dimension |
| Each interval order can be embedded into interval orders of sufficiently large dimension. More precisely, if X is an interval order, there exists a number t = t(X), such that for every interval order Y of dimension at least t, Y contains a subposet isomorphic to X. This is proven by embedding interval orders into some fixed structure, called "a thicket", and then finding thickets in all orders of sufficiently large dimension. |
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22.07.2005 Ralph McKenzie,Vanderbilt University, Nashville TN, USA |
Informatyka Teoretyczna Operations on finite structures |
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01.07.2005 Nobu-Yuki Suzuki,Shizuoka University, Japan |
Informatyka Teoretyczna Epistemic logic and game theory |
| A brief overview of epistemic logics and their game-theoretical applications is given. This interdisciplinary field has many aspects arising from various research lines. In this talk, I give an outline of the recent developments of epistemic-logical approach to decision making process in game-theoretical situations. |
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01.06.2005 Grzegorz Łukasik |
Informatyka Teoretyczna Mobile Robots Contests |
| Mobile Robots Contests organized by Computer Science and Technology University is presented: rules of the contest, the problems that students were solving and intresting solutions that were implemented. The first three places were taken by teams from Jagiellonian University. References:Contest home page: http://www.best.agh.edu.pl/konkurs/ |
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18.05.2005 Michael Sołtys,McMaster University,Hamilton, ON, Canada |
Informatyka Teoretyczna P vs NP |
| The "P vs NP" problem is a central problem of theoretical computer science, and indeed of mathematics (it was named one of the seven Millennium Problem by the Clay Mathematical Institute, and there is a one million dollar prize for a solution: http://www.claymath.org/millennium). Despite thirty years of intense efforts, we are not near a solution, and we do not have promising techniques to tackle this problem. For example, since diagonal arguments "relativize", they will probably not work in this case. Exponential lower bounds on Boolean circuits are also hopelessly difficult to obtain. De-randomizing turns out to be just as difficult as separating P and NP. Cook proposed an attack based on a program of finding lower bounds for stronger and stronger propositional proof systems, building a repertoire of techniques and lower bounds, and ultimately showing that no polynomially bounded propositional proof system exists. The consequence of that would be that NP is not equal to coNP, and thus P is not equal to NP. In this talk, I will concentrate on this line of attack. |
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16.05.2005 11.05.2005,Lech Palmowski |
Informatyka Teoretyczna Seventeen lines and one-hundred-and-one points |
| A curious problem from additive number theory is investigated: Given two positive integers S, Q, does there exist a sequence of positive integers that add up to S and whose squares add up to Q? We show that this problem can be solved in time polynomially bounded in the logarithms of S and Q. As a consequence, also the following question can be answered in polynomial time: For given numbers n and m, do there exist n lines in the Euclidean plane with exactly m points of intersection? References:G. J. Woeginger, Seventeen lines and one-hundred-and-one points, Theoretical Computer Science 321 (2004), 415-421 |
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04.05.2005 27.04.2005 Grzegorz Gutowski |
Informatyka Teoretyczna Computational aspects of the 2-dimension of partially ordered sets |
A well-known method to represent a partially ordered set consists in associating to each element of P a subset of a fixed set S={1,..,k} such that the order relation coincides with subset inclusion. Given an order P, minimizing the size of the encoding, i.e. the cardinal of S, is however a difficult problem. The smallest size is called the 2-dimension of P. The paper details a proof that computing 2-dimension of a given poset is NPC. Reduction allows to prove the non-approximability of the problem. Later on the complexity of the 2-dimension for the class of trees is investigated. Authors present a 4-approximation algorithm for this class. |
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20.04.2005 30.03.2005 Grzegorz Herman |
Informatyka Teoretyczna The Complexity of Random Ordered Structure |
One of the ways of expressing the complexity of a structure is the minimal "depth" of quantifiers in a formula that describes it. The presented paper discusses such complexity of two types of structures. The authors prove that the complexity of a random bit string is O(loglogn), with high probability, and the complexity of an ordered random graph - Theta(log*n), with high probability. |
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23.03.2005 Mikołaj Zalewski |
Informatyka Teoretyczna On-line Ramsey Theory |
| The Ramsey game is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monchromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the winner for a given pair H, G. In particular, authors of paper prove that Builder has a winning strategy for any k-colorable graph H in the game played on k-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. They show that the class of outerplanar graphs does not have this proberty. The question of whether planar graphs are self-unaviodable is left open. References:J.A. Grytczuk, M. Haluszczak, H.A.Kierstead, On-line Ramsey Theory, The Elektronic Journal of Combinatorics 11(1), 2004 |
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09.03.2005 Wiktor Żelazny |
Informatyka Teoretyczna Online Algorithms for Market Clearing |
| Randomized on-line algorithms that maximalize profit and liquidity of marketplace are presented. They work by finding a maximal set with perfect matching in subgraph of bid history graph - before incoming intervals are applied, some of them (or some of graph edges from incoming vertices) are removed, based at random criteria. Proof that estimated profit produced by optimal algorithm cannot be greater then estimated profit of presented algorithm was also shown. References:A. Blum, T. Sandholm, M. Zinkevich, Online Alghoritms for Market Clearing, manuscript (2002) |
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01.03.2005 Wiktor Żelazny |
Informatyka Teoretyczna Online Algorithms for Market Clearing |
| This talk presents an online algorithm able to find maximal set W of vertexes of n-interval graph, such that a perfect matching exists on W. The alghoritm works on interval representation of graph and new intervals must be introduced in legal way (described in previous talk) for it to work, but it's competitive ratio is equal 1. Proof of alghoritm's optimality was presented. References:A. Blum, T. Sandholm, M. Zinkevich, Online Alghoritms for Market Clearing, manuscript (2002) |
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23.02.2005 Jacek Krzaczkowski |
Informatyka Teoretyczna Counting solutions of equations over two-element algebras. |
| I refered my results connected with counting solutions of equations and systems of equations over fixed two-element algebra. It came out that the computational complexity of these problems depends only on termal clone of the algebra. I presented the classification of the computational complexity of the problems mentioned above. |
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26.01.2005 Wiktor Żelazny |
Informatyka Teoretyczna Online Algorithms for Market Clearing |
| A n-interval graph is created from interval graph by painting it's vertexes with n colours and removing all edges between vertices of the same colour. This talk introduced n-interval graphs, as well as the way their interval representation can represent history of buy and sell bids on exchange auction. Objectives of maximalizing marketplace's profit and liquidity are formulated as online problems, as new intervals are being introduced every time a new bid appears. A way of legal introduction of new intervals, corresponding to apperance of new bids, is described. References:A. Blum, T. Sandholm, M. Zinkevich, Online Alghoritms for Market Clearing, manuscipt (2002) |
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12.01.2005 Grzegorz Gronkowski |
Informatyka Teoretyczna A boolean function requiring 3n network size. |
| First part of talk contains a simple proof, that there exist functions with a nonlinear lower bound for the network complexity. The proof is based on a counting method. Afterwards I present Norbert Bloom's result, who defined n-ary boolean function with a 3n lower bound. The proof shows, that 3n-3 is a minimal number of logic gates which is necessary for computing this function. References:Norbert Bloom, A boolean function requiring 3n network size. |
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05.01.2005 T. Krawczyk,E. Szczypka |
Informatyka Teoretyczna Comparability graphs. |
| A graph G=(V,E) is a comparability graph if there exists a partial order (V,<) such that there exists an edge between vertices v and w iff either v < w or w < v. In our talk we presented a polynomial time algorithm (the best known that solves a similar problem - Spinrad, McConnell - has a complexity O(n+m)) for deciding whether a given graph G=(V,E) is a comparability graph. In our method we investigated relations that exist between sets of neighborhoods in comparability graphs. |
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15.12.2004 Marek Kwiatkowski |
Informatyka Teoretyczna Dr. Frankenstein's Approach to On-line Algorithms |
| Let A1...An be on-line algorithms for a problem P, competitive over subsets I1...In of inputs respectively. In this talk we show that under certain assumptions on P and I1...In it is possible to give an on-line algorithm for P that is competitive over all possible inputs. Two solutions are given: for deterministic and randomized algorithms. References:Yossi Azar, Andrei Z. Broder, Mark S. Manasse "Dr. Frankenstein's Approach to On-line Algorithms" (extended abstract) |
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08.12.2004 Krzysztof Maczyński |
Informatyka Teoretyczna P-time algorithm for tree decomposition |
| In my talk I presented a polynomial time algorithm for deciding whether a given tree is arbitrarily vertex decomposable (AVD) in a limited sense concerning no more than a constant number of components and if so, constructing one of possibly exponentially many such decompositions. I also assumed a constant bound on the maximal degree of the tree but this condition was shown to be unnecessary by a slight modification to my algorithm proposed by Mikołaj Zalewski. Additionally I detailed a construction of a maximally long greedy sequence over a set of colours whose cardinality is given. I characterized all sequences of equal length, proved that no longer ones exist and explicited a cubic function computing the length from the number of colours allowed. |
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01.12.2004 Jan Jeżabek |
Informatyka Teoretyczna Graphs with high girth and high chromatic number |
| This talk introduces a basic tool of the probabilistic method - the first moment method. The method is illustrated with an application to satisfiability problems. A simple theorem is presented stating that any instance of k-SAT with fewer than 2^k clauses is satisfiable. The first moment method is then used to prove that for every g, k >= 1 there exist graphs with no cycles of size g and with chromatic number greater than k. References:M.Molloy, The Probabilistic Method, in: M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, B. Reed, Probabilistic Methods for Algorithmic Discrete Mathematics, Springer 1998 |
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24.11.2004 Piotr Micek |
Informatyka Teoretyczna Natural algorithm for Online Chain Covering of Upgrowing Interval Posets |
| We have already proven that a competitive ratio for Online Chain Covering of Upgrowing Posets is equal to 2. At this talk I present a simple online algorithm which has optimal competitive ratio. |
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17.11.2004 10.11.2004 Bartłomiej Bosek |
Informatyka Teoretyczna Lowerbound for Online Chain Partitioning of Upgrowing Interval Posets |
Bartek presents that there is no online algorithm for chain covering problem of upgrowing posets using less than 2w-1 chains to cover a width w poset. |
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03.11.2004 27.10.2004 Paweł Walter |
Informatyka Teoretyczna Online Bin Packing with two item sizes |
In the well-studied on-line bin packing problem (BPP) we are given a set of items and a sequence of their sizes and are required to pack them into a minimum number of unit-capacity bins. The problem of finding the competitive ratio in the general case is open. In the analysed paper BPP is considered restricted to the case of two distinct item sizes. My talk based on this paper consists of an algorithm solving this particular case at an asymptotic competetive ratio of 4/3 and a proof that 4/3 is also here a valid lower bound. |
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13.10.2004 Tomek Krawczyk |
Informatyka Teoretyczna NP-completeness of posets embedding into boolean lattice |
| Let p < q and p,q from N. A bipartitie graph G=(X \cup Y,E) embeds into a lattice of subsets on levels p and q if there exist n from N, injections f: X --> {n \choose p}, g: Y--> {n \choose q} such that there is an edge between x from X and y from Y iff f(x) < g(y). In my talk I proved that the problem of deciding whether a given bipartite graph G=(X \cup Y,E) embeds into lattice of subsets on levels p and q is NP-complete if p>1 and is in P if p=1. |
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