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Theoretical computer science
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Poprzednie referaty

25.01.2024 17:30
Filip Jasionowicz
Combinatorial Optimization
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.

  1. Michael A. Bekos, Michael Kaufmann, Fabian Klute, Sergey Pupyrev, Chrysanthi Raftopoulou, Torsten Ueckerdt. Four Pages Are Indeed Necessary for Planar Graphs. arXiv:2004.07630. (2020).
  2. Filip Jasionowicz. Book embeddings of graphs and why four pages are indeed necessary for planar graphs. slides. (2024).
25.01.2024 16:45
Artemy Oueiski
Combinatorial Optimization
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.

  1. D.G. Corneil, H. Lerchs, L.Stewart Burlingham. Complement reducible graphs. Discrete Applied Mathematics. 3(3), 163-174. (1981).
  2. Hans L. Bodlaender and Rolf H. Möhring. The Pathwidth and Treewidth of Cographs. SIAM Journal on Discrete Mathematics. 6(2). 181-188. (1993).
  3. Artemy Oueiski. A simple linear time algorithm for cograph recognition. slides. (2024).
25.01.2024 16:00
Katzper Michno
Combinatorial Optimization
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.

  1. Matthieu Rosenfeld. Another approach to non-repetitive colorings of graphs of bounded degree. arXiv:2006.09094. (2020).
  2. Katzper Michno. Another approach to non-repetitive colorings of graphs of bounded degree. slides. (2024).
24.01.2024 16:15
Sergio Cabello
University of Ljubljana and IMFM, Slovenia
Theoretical computer science
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.

18.01.2024 16:45
Milana Kananovich
Combinatorial Optimization
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).

  1. D. G. Corneil, Y. Perl, and L. K. Stewart. A Linear Recognition Algorithm for Cographs. SIAM Journal on Computing. 14(4). 926-934. (1985).
  2. Anna Bretscher, Derek Corneil, Michel Habib, and Christophe Paul. A Simple Linear Time LexBFS Cograph Recognition Algorithm. SIAM Journal on Discrete Mathematics. 22(4). 1277-1296. (2008).
  3. Milana Kananovich. Cograph Recognition. slides. (2024).
18.01.2024 16:00
Sebastian Spyrzewski
Combinatorial Optimization
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.

  1. Zdeněk Dvořák, Sergey Norin, Luke Postle. List coloring with requests. arXiv:1612.08698. (2016).
  2. Sebastian Spyrzewski. List coloring with requests. slides. (2024).
17.01.2024 16:15
Jim Geelen
University of Waterloo
Theoretical computer science
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.

11.01.2024 16:45
Aleksander Katan
Combinatorial Optimization
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.

  1. John Haslegrave. Countable graphs are majority 3-choosable. arXiv:2003.10408. (2020).
  2. Aleksander Katan. Countable graphs are majority 3-choosable. slides. (2024).
11.01.2024 16:00
Łukasz Gniecki
Combinatorial Optimization
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.

  1. Yangyan Gu, Xuding Zhu. The Alon-Tarsi number of planar graphs - a simple proof. arXiv:2203.16308. (2022).
  2. Łukasz Gniecki. The Alon Tarsi Number of Planar Graphs - a Simple Proof. slides. (2024).
10.01.2024 16:15
Peter Allen
London School of Economics and Political Science
Theoretical computer science
Universality for degenerate graphs

A graph G is universal for a family of graphs if for each F in  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 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.

21.12.2023 16:45
Kamil Galewski
Combinatorial Optimization
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 d< 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.

  1. Dan Hefetz. On two generalizations of the Alon-Tarsi polynomial method. arXiv:0911.2099. (2009).
  2. Kamil Galewski. Generalizations of the Alon-Tarsi polynomial method. slides. (2023).
21.12.2023 16:00
Ignacy Buczek
Combinatorial Optimization
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.

  1. Ignacy Buczek. A note on computer-assisted proofs in flag algebras. slides. (2023).
20.12.2023 16:15
Paweł Rzążewski
Warsaw University of Technology
Theoretical computer science
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.

14.12.2023 16:00
Jędrzej Hodor
Combinatorial Optimization
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.

  1. W. A. Wythoff. A modification of the game of nim. Nieuw Archief voor Wiskunde. 2(7). 199-202. (1907).
  2. Jędrzej Hodor. Wythoff’s game. slides. (2023).
13.12.2023 00:00

Theoretical computer science
The 17th workshop on Computational Logic and Applications
07.12.2023 16:00
Piotr Kaliciak
Combinatorial Optimization
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.

  1. Charlotte Knierim, Anders Martinsson, Raphael Steiner. Hat guessing numbers of strongly degenerate graphs. arXiv:2112.09619. (2021).
  2. Piotr Kaliciak. Hat guessing numbers of strongly degenerate graphs. slides. (2023).
06.12.2023 16:15
Paul Bastide
LaBRI, Bordeaux
Theoretical computer science
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.

30.11.2023 16:45
Jan Klimczak
Combinatorial Optimization
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.

  1. Hamza B. Habib. On the equitable distribution of points on the circle. MSc thesis. University of Wollongong. (2014).
  2. Jan Klimczak. On the equitable distribution of points on the circle. slides. (2023).
30.11.2023 16:00
Rafał Pyzik
Combinatorial Optimization
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.

  1. Niv Buchbinder, D. Segev, Yevgeny Tkach. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals. Algorithmica. 81, 1781-1799. (2019)
  2. Rafał Pyzik. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals slides. (2023).
29.11.2023 16:15
Matthieu Rosenfeld
LIRMM, Montpellier
Theoretical computer science
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.

In this presentation, I will present this counting argument. We will apply this technique to one or two simple examples from graph theory to illustrate how it works. I will try to give a broad intuition behind a couple of more involved results that were achieved by different authors with this technique.

23.11.2023 16:45
Izabela Tylek
Combinatorial Optimization
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.

  1. Izabela Tylek. Any 7-chromatic graph has K7 or K4,4 as a minor. slides. (2023).
23.11.2023 16:00
Justyna Jaworska
Combinatorial Optimization
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.

  1. Wei-Kuan Shih, Wen-Lian Hsu. An O(n1.5) algorithm to color proper circular arcs. Discrete Applied Mathematics. 25(3), 321-323. (1989).
  2. Justyna Jaworska. An O(n1.5) algorithm to color proper circular arcs. slides. (2023).
22.11.2023 16:15
Gábor Damásdi
Alfréd Rényi Institute of Mathematics
Theoretical computer science
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  d1d2≥...≥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 k7. 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.

 

16.11.2023 16:45
Maciej Sanocki
Combinatorial Optimization
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.

  1. Maciej Sanocki. Two-sided Online Bipartite Matching and Vertex Cover: Beating the Greedy Algorithm. slides. (2023).
16.11.2023 16:00
Katarzyna Kępińska
Combinatorial Optimization
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.

  1. Katarzyna Kępińska. On Two problems of Defective Choosability of Graphs. slides. (2023).
15.11.2023 16:15
Piotr Micek
Jagiellonian
Theoretical computer science
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

09.11.2023 16:00
Karolina Okrasa
Warsaw University of Technology
Combinatorial Optimization
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.

08.11.2023 16:15
Torsten Ueckerdt
Karlsruhe Institute of Technology
Theoretical computer science
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.

26.10.2023 16:45
Agata Margas
Combinatorial Optimization
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.

  1. Agata Margas. Making the Rules of Sports Fairer. slides. (2023).
26.10.2023 16:00
Mikołaj Kot
Combinatorial Optimization
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.

  1. Bruno Courcelle. Circle graphs and monadic second-order logic. Journal of Applied Logic. 6(3). 416-442. (2008).
  2. Mikołaj Kot. Circle graphs and monadic second-order logic. slides. (2023).
26.10.2023 14:15
Jan Klimczak, Szymon Wojtulewicz
Algorytmika
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.
25.10.2023 16:15
Torsten Mütze
University of Warwick
Theoretical computer science
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 n1.

19.10.2023 16:45
Maksym Zub
Combinatorial Optimization
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.

  1. Manouchehr Zaker. A note concerning the Grundy and b-chromatic number of graphs. arXiv :2003.14233. (2020).
  2. Maksym Zub. A note concerning the Grundy and b-chromatic number of graphs. slides. (2023).
19.10.2023 16:00
Jakub Fedak
Combinatorial Optimization
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).

  1. M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou. SIAM Journal on Algebraic Discrete Methods. 1(2). 216-227. (1980).
  2. Jakub Fedak. The Complexity of Coloring Circular Arcs and Chords. slides. (2023).
18.10.2023 16:15
Marcelo Campos
University of Oxford
Theoretical computer science
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.


This is joint work with Simon Griffiths, Robert Morris, Julian Sahasrabudhe.

12.10.2023 16:00
Bartłomiej Błoniarz
Combinatorial Optimization
(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.

  1. Bartłomiej Błoniarz. (Some of) the many uses of Eulerian graphs in graph theory (plus some applications). slides. (2023).
  2. H. Fleischner. (Some of) the many uses of Eulerian graphs in graph theory (plus some applications) Discrete Mathematics 230(1-3). 23-43. (2001).
12.10.2023 14:15
Kacper Topolski, Jakub Wąs
Algorytmika
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.
12.10.2023 14:15
Kacper Topolski, Jakub Wąs
Algorytmika
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.
11.10.2023 16:15
Krzysztof Potępa
Jagiellonian University
Theoretical computer science
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.

05.10.2023 16:00
Bartłomiej Bosek
Combinatorial Optimization
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.

04.10.2023 16:15
Avi Wigderson
Institute for Advanced Study, Princeton
Theoretical computer science
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/2001.04383

As it happens, both acronyms MIP* and RE represent proof systems, of a very different nature. To explain them, we'll take a meandering jurney through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (of both problems and proofs) by algorithmic efficiency, naturally leads to the genaration of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties.

The talk does not require special mathematical background.

15.06.2023 16:45
Julia Biały
Combinatorial Optimization
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.

  1. Julia Biały. A game generalizing Hall's Theorem. slides. (2023).
  2. Landon Rabern. A game generalizing Hall's theorem. arXiv:1204.0139. (2012).
15.06.2023 16:00
Łukasz Selwa
Combinatorial Optimization
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.

  1. Michael Stiebitz, Zsolt Tuza, Margit Voigt. Orientations of Graphs with Prescribed Weighted Out-Degrees. Graphs and Combinatorics 31, 265-280. (2015).
14.06.2023 16:15
Fabrizio Frati
Università Roma Tre
Theoretical computer science
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.

07.06.2023 16:15
Michał Seweryn
Université Libre de Bruxelles
Theoretical computer science
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

31.05.2023 16:15
Ola Svensson
École Polytechnique Fédérale de Lausanne
Theoretical computer science
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
25.05.2023 16:45
Katarzyna Król
Combinatorial Optimization
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.

  1. Katarzyna Król. Ball Packing. slides. (2023).
  2. Hao Chen, Jean-Philippe Labbé. Lorentzian Coxeter systems and Boyd-Maxwell ball packings. arXiv:1310.8608. (2012).
25.05.2023 16:00
Jędrzej Kula
Combinatorial Optimization
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.

  1. Jędrzej Kula. Playing cards with Vizing's demon. slides. (2023).
  2. Brian Rabern, Landon Rabern. Playing cards with Vizing's demon. arXiv:2104.04624. (2021).
24.05.2023 16:15
Csaba Tóth
California State University, Northridge
Theoretical computer science
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.

18.05.2023 16:45
Krzysztof Barański
Combinatorial Optimization
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.

  1. Krzysztof Barański. A note on degree-constrained subgraphs. slides. (2023).
  2. András Frank, Lap Chi Lau, Jácint Szabó. A note on degree-constrained subgraphs. Discrete Mathematics. 308(12) 2647-2648. (2008).
18.05.2023 16:00
Filip Konieczny
Combinatorial Optimization
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.

  1. Filip Konieczny. On constructive methods in the theory of colour-critical graphs. slides. (2023).
  2. Horst Sachs, Michael Stiebitz. On constructive methods in the theory of colour-critical graphs. Discrete Mathematics. 74(1-2), 201-226. (1989).
17.05.2023 16:15
John Iacono
Université Libre de Bruxelles
Theoretical computer science
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.

11.05.2023 16:45
Rafał Pyzik
Combinatorial Optimization
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.

  1. Rafał Pyzik. Improved lower bounds on the number of edges in list critical and online list critical graphs. slides. (2023).
  2. H.A. Kierstead, Landon Rabern. Improved lower bounds on the number of edges in list critical and online list critical graphs. Journal of Combinatorial Theory, Series B. 140, 147-170. (2020).
11.05.2023 16:00
Aleksander Katan
Combinatorial Optimization
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.

  1. Aleksander Katan. A not 3-choosable planar graph without 3-cycles. slides. (2023).
  2. Margit Voigt. A not 3-choosable planar graph without 3-cycles. Discrete Mathematics. 146(1-3), 325-328. (1995).
  3. C. Thomassen. Every Planar Graph Is 5-Choosable. Journal of Combinatorial Theory, Series B. 62(1), 180-181. (1994).
10.05.2023 16:15
Clément Rambaud
École Normale Supérieure, PSL Paris
Theoretical computer science
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.

04.05.2023 16:45
Rafał Kilar
Combinatorial Optimization
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.

  1. Rafał Kilar. On the structure of k-connected graphs without Kk-minor. slides. (2023).
  2. Ken-ichi Kawarabayashi, Rong Luo, Jianbing Niu, Cun-Quan Zhang. On the structure of k-connected graphs without Kk-minor. European Journal of Combinatorics. 26(3-4), 293-308. (2005).
04.05.2023 16:00
Bartłomiej Błoniarz
Combinatorial Optimization
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.

  1. Bartłomiej Błoniarz. Pólya's Permanent Problem. slides. (2023).
  2. William McCuaig. Pólya's Permanent Problem. Electronic Journal of Combinatorics. 11(1), R79. (2004).
27.04.2023 16:45
Demian Banakh
Combinatorial Optimization
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.

  1. Demian Banakh. Flip distance to a non-crossing perfect matching. slides. (2023).
  2. Édouard Bonnet, Tillmann Miltzow. Flip Distance to a Non-crossing Perfect Matching. arXiv:1601.05989. (2016).
  3. Guilherme D. da Fonseca, Yan Gerard, Bastien Rivier. On the Longest Flip Sequence to Untangle Segments in the Plane. arXiv:2210.12036. (2022).
27.04.2023 16:00
Szymon Salabura
Combinatorial Optimization
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.

  1. Szymon Salabura. Edge lower bounds for list critical graphs, via discharging. slides. (2023).
  2. Daniel W. Cranston, Landon Rabern. Edge Lower Bounds for List Critical Graphs, via Discharging. arXiv:1602.02589. (2016).
27.04.2023 14:15
Justyna Jaworska, Jakub Siuta
Algorytmika
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.
26.04.2023 16:15
David Eppstein
University of California, Irvine
Theoretical computer science
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.

20.04.2023 16:45
Piotr Kaliciak
Combinatorial Optimization
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.

  1. Piotr Kaliciak. Decomposing 4-connected planar triangulations into two trees and one path. slides. (2023).
  2. Kolja Knauer, Torsten Ueckerdt. Decomposing 4-connected planar triangulations into two trees and one path. Journal of Combinatorial Theory, Series B. 134, 88-109. (2019).
20.04.2023 16:00
Kamil Galewski
Combinatorial Optimization
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.

  1. Kamil Galewski. On the discrepancy of circular sequences of reals. slides. (2023).
  2. Fan Chung, Ron Graham. On the discrepancy of circular sequences of reals. Journal of Number Theory. 164, 52-65. (2016).
19.04.2023 16:15
Pat Morin
Carleton University
Theoretical computer science
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 ts, 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.

13.04.2023 16:45
Łukasz Gniecki
Combinatorial Optimization
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.

  1. Łukasz Gniecki. Sequences of points on a circle. slides. (2023).
  2. N.G. de Bruijn, P. Erdös. Sequences of points on a circle. Proceedings of the Section of Sciences of the Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam. 52(1):14-17. (1949).
13.04.2023 16:00
Ignacy Buczek
Combinatorial Optimization
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.

  1. Ignacy Buczek. 10 Problems for Partitions of Triangle-free Graphs. slides. (2023).
  2. József Balogh, Felix Christian Clemen, Bernard Lidický. 10 Problems for Partitions of Triangle-free Graphs. arXiv:2203.15764. (2022).
13.04.2023 14:15
Rafał Pyzik, Sebastian Spyrzewski
Algorytmika
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.
12.04.2023 16:15
Ruta Mehta
University of Illinois at Urbana-Champaign
Theoretical computer science
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.

05.04.2023 16:15
Ralph Keusch
Siemens Mobility CH
Theoretical computer science
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.

30.03.2023 16:45
Jakub Siuta
Combinatorial Optimization
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.

  1. Jakub Siuta. List-avoiding orientations. slides. (2023).
  2. Peter Bradshaw, Yaobin Chen, Hao Ma, Bojan Mohar, Hehui Wu. List-avoiding orientations. arXiv:2209.09107. (2022).
30.03.2023 16:00
Grzegorz Gawryał
Combinatorial Optimization
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.

  1. Grzegorz Gawryał. Critically paintable, choosable or colorable graphs. slides. (2023).
  2. Ayesha Riasat, Uwe Schauz. Critically paintable, choosable or colorable graphs. Discrete Mathematics. 312 (22), 3373-3383. (2012).
30.03.2023 14:15
Łukasz Grobelczyk, Rafał Loska
Algorytmika
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.
29.03.2023 16:15
Alex Scott
University of Oxford
Theoretical computer science
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.

23.03.2023 16:45
Tomasz Mazur
Combinatorial Optimization
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.

  1. Tomasz Mazur. A note on large induced subgraphs with prescribed residues in bipartite graphs. slides. (2023).
  2. Zach Hunter. A note on large induced subgraphs with prescribed residues in bipartite graphs. arXiv:2201.00296. (2022).
23.03.2023 16:00
Katzper Michno
Combinatorial Optimization
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.

  1. Katzper Michno. Dimension and cut vertices: an application of Ramsey theory. slides. (2023).
  2. William T. Trotter, Bartosz Walczak, Ruidong Wang. Dimension and cut vertices: an application of Ramsey theory. arXiv:1505.08162. (2015).
22.03.2023 16:15
Martin Grohe
RWTH Aachen
Theoretical computer science
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.