Discrete Mathematics Group

Jaehoon Kim (김재훈), Ramsey numbers of cycles versus general graphs
2022-03-21 16:30 - 17:30 (Asia/Seoul) | Room B232, IBS (기초과학연구원)
The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(chi(H)-1)+sigma(H)$, where $sigma(H)$ is the minimum possible size of a colour class in a $chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $ngeq |H|chi(H)$.
We improve this 40-year-old result of Burr by giving quantitative bounds of the form $ngeq C|H|log^4chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number.
This is joint work with John Haslegrave, Joseph Hyde and Hong Liu

Ben Lund, Thresholds for incidence properties in finite vector spaces
2022-03-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Suppose that $E$ is a subset of $mathbb{F}_q^n$, so that each point is contained in $E$ with probability $theta$, independently of all other points. Then, what is the probability that there is an $m$-dimensional affine subspace that contains at least $ell$ points of $E$? What is the probability that $E$ intersects all $m$-dimensional affine subspaces? We give Erdős-Renyi threshold functions for these properties, in some cases sharp thresholds. Our results improve previous work of Chen and Greenhill. This is joint work with Jeong Han Kim, Thang Pham, and Semin Yoo.

Jean-Florent Raymond, Long induced paths in minor-closed graph classes and beyond
2022-03-30 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
In 1982 Galvin, Rival, and Sands proved that in $K_{t,t}$-subgraph free graphs (t being fixed), the existence of a path of order n guarantees the existence of an induced path of order f(n), for some (slowly) increasing function f. The problem of obtaining good lower-bounds for f for specific graph classes was investigated decades later and logarithmic bounds have been obtained for planar graphs (more generally for graphs of bounded genus) and for interval graphs.
In this talk I will show that every graph of pathwidth less than k that has a path of order n also has an induced path of order $Ω(n^{1/k})$. I will then explain how this result can be used to prove the two following generalizations:

every graph of treewidth less than k that has a path of order n contains an induced path of order $Ω((log n)^{1/k})$;
for every non-trivial graph class that is closed under topological minors there is a constant d∈(0,1) such that every graph from this class that has a path of order n contains an induced path of order $Ω((log n)^d)$.

Joint work with Claire Hilaire.

Younjin Kim (김연진), On the extremal problems related to Szemerédi's theorem
2022-04-11 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In 1975, Szemerédi proved that for every real number $delta > 0 $ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set ${1, 2, cdots, N }$ with $|A| geq delta N$ contains an arithmetic progression of length $k$. There has been a plethora of research related to Szemerédi’s theorem in many areas of mathematics. In 1990, Cameron and Erdős proposed a conjecture about counting the number of subsets of the set ${1,2, dots, N}$ which do not contain an arithmetic progression of length $k$. In the talk, we study a natural higher dimensional version of this conjecture, and also introduce recent extremal problems related to Szemerédi’s theorem.

Jakub Gajarský, Model Checking on Interpretations of Classes of Bounded Local Clique-Width
2022-04-13 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The first-order model checking problem for finite graphs asks, given a graph G and a first-order sentence $phi$ as input, to decide whether $phi$ holds on G. Showing the existence of an efficient algorithm for this problem implies the existence of efficient parameterized algorithms for various commonly studied problems, such as independent set, distance-r dominating set, and many others.
While the first-order model-checking problem is likely not efficiently solvable in general, efficient algorithms exist for various restricted graph classes, such as graphs of bounded degree, planar graphs etc. After the existence of an efficient model checking algorithm was shown for nowhere dense classes of graphs (which include most of commonly studied classes of sparse graphs), the attention turned to the more general setting of graph classes which can be obtained from sparse graphs using graph transformations called interpretations/transductions. However, despite efforts of several groups of researchers, no positive algorithmic result has been achieved since 2016, when the existence of an efficient algorithm was shown for graph classes interpretable in graphs of bounded degree.
We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local clique-width. Notably, this includes interpretations of planar graphs (and more generally, of locally bounded treewidth) and vastly generalizes the result for interpretations of graphs of bounded degree. To obtain this result we developed a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future.
This is joint work with Édouard Bonnet, Jan Dreier, Stephan Kreutzer, Nikolas Mählmann, Pierre Simon, Szymon Toruńczyk.

Boram Park (박보람), Odd coloring of sparse graphs
2022-04-25 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We introduce an odd coloring of a graph, which was introduced very recently, motivated by parity type colorings of graphs. A proper vertex coloring of graph $G$ is said to be odd if for each non-isolated vertex $x in V (G)$ there exists a color $c$ such that $c$ is used an odd number of times in the neighborhood of $x$. The recent work on this topic will be presented, and the work is based on Eun-Kyung Cho, Ilkyoo Choi, and Hyemin Kown.

Michael Savery, Induced subgraphs of induced subgraphs of large chromatic number
2022-04-27 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
We prove that for every graph F with at least one edge 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 c=c(F). (Here a graph is F-free if it does not contain an induced copy of F.) This generalises recent 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. This is joint work with Antonio Girão, Freddie Illingworth, Emil Powierski, Alex Scott, Youri Tamitegama and Jane Tan.

Cheolwon Heo (허철원), The complexity of the matroid-homomorphism problems
2022-05-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In this talk, we introduce homomorphisms between binary matroids that generalize graph homomorphisms. For a binary matroid $N$, we prove a complexity dichotomy for the problem $rm{Hom}_mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial-time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.
This is joint work with Hyobin Kim and Mark Siggers at Kyungpook National University.

Kyeongsik Nam (남경식), Large deviations for subgraph counts in random graphs
2022-05-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The upper tail problem for subgraph counts in the Erdos-Renyi graph, introduced by Janson-Ruciński, has attracted a lot of attention. There is a class of Gibbs measures associated with subgraph counts, called exponential random graph model (ERGM). Despite its importance, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In this talk, I will talk about a brief overview on the upper tail problem and the concentration of measure results for the ERGM. Joint work with Shirshendu Ganguly and Ella Hiesmayr.

Andreas Holmsen, A colorful version of the Goodman-Pollack-Wenger transversal theorem
2022-05-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Hadwiger’s transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in $mathbb{R}^d$ by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of Arocha, Bracho, and Montejano. The proof uses topological methods, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.

Jan Kurkofka, Canonical Graph Decompositions via Coverings
2022-05-18 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed.
The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure at the intended level of locality while unfolding its global structure.
Our theorem extends to locally finite quasi-transitive graphs and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition theorem for finitely generated groups into local parts, whose relative structure is displayed by a graph.
Joint work with Reinhard Diestel, Raphael W. Jacobs and Paul Knappe.

Gil Kalai, The Cascade Conjecture and other Helly-type Problems
2022-05-19 16:15 - 17:15 | Zoom ID: 868 7549 9085
[Colloquium, Department of Mathematical Sciences, KAIST]
For a set $X$ of points $x(1)$, $x(2)$, $ldots$, $x(n)$ in some real vector space $V$ we denote by $T(X,r)$ the set of points in $X$ that belong to the convex hulls of r pairwise disjoint subsets of $X$.
We let $t(X,r)=1+dim(T(X,r))$.
Radon’s theorem asserts that
If $t(X,1)< |X|$, then $t(X, 2) >0$.
The first open case of the cascade conjecture asserts that
If $t(X,1)+t(X,2) < |X|$, then $t(X,3) >0$.
In the lecture, I will discuss connections with topology and with various problems in graph theory. I will also mention questions regarding dimensions of intersection of convex sets.
Some related material:
1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos
2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804
3) A link to Barany’s book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097

Stijn Cambie, The precise diameter of reconfiguration graphs
2022-05-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Reconfiguration is about changing instances in small steps. For example, one can perform certain moves on a Rubik’s cube, each of them changing its configuration a bit. In this case, in at most 20 steps, one can end up with the preferred result. One could construct a graph with as nodes the possible configurations of the Rubik’s cube (up to some isomorphism) and connect two nodes if one can be obtained by applying only one move to the other. Finding an optimal solution, i.e. a minimum number of moves to solve a Rubik’s cube is now equivalent to finding the distance in the graph.
We will wonder about similar problems in reconfiguration, but applied to list- and DP-colouring. In this case, the small step consists of recolouring precisely one vertex. Now we will be interested in the diameter of the reconfiguration graph and show that sometimes we can determine the precise diameters of these.
As such, during this talk, we present some main ideas of [arXiv:2204.07928].

Sebastian Siebertz, Transducing paths in graph classes with unbounded shrubdepth
2022-05-25 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class C can be FO-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from C one cannot FO-transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the MSO-transduction quasi-order, even in the stronger form that concerns FO-transductions instead of MSO-transductions.
The backbone of our proof is a graph-theoretic statement that says the following: If a graph G excludes a path, the bipartite complement of a path, and a half-graph as semi-induced subgraphs, then the vertex set of G can be partitioned into a bounded number of parts so that every part induces a cograph of bounded height, and every pair of parts semi-induce a bi-cograph of bounded height. This statement may be of independent interest; for instance, it implies that the graphs in question form a class that is linearly chi-bounded.
This is joint work with Patrice Ossona de Mendez and Michał Pilipczuk.

Hongseok Yang (양홍석), Learning Symmetric Rules with SATNet
2022-05-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given as images. In this paper, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik’s cube show the substantial improvement of SymSATNet over the baseline SATNet.
This is joint work with Sangho Lim and Eungyeol Oh.

Jeck Lim, Sums of linear transformations
2022-06-02 10:30 - 11:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
We show that if $L_1$ and $L_2$ are linear transformations from $mathbb{Z}^d$ to $mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $mathbb{Z}^d$, [ |L_1 A+L_2 A|geq (|det(L_1)|^{1/d}+|det(L_2)|^{1/d})^d |A|- o(|A|).] This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for many choices of $L_1$ and $L_2$. As an application, we prove a lower bound for $|A + lambda cdot A|$ when $A$ is a finite set of real numbers and $lambda$ is an algebraic number.
Joint work with David Conlon.

O-joung Kwon (권오정), Graph minor theory and beyond
2022-06-02 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
[Colloquium, Department of Mathematical Sciences, KAIST]
One of the important work in graph theory is the graph minor theory developed by Robertson and Seymour in 1980-2010. This provides a complete description of the class of graphs that do not contain a fixed graph H as a minor. Later on, several generalizations of H-minor free graphs, which are sparse, have been defined and studied. Also, similar topics on dense graph classes have been deeply studied. In this talk, I will survey topics in graph minor theory, and discuss related topics in structural graph theory.

Amadeus Reinald, Twin-width and forbidden subdivisions
2022-06-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse, notably including bounded rank-width, $Omega ( log (n) )$-subdivisions of graphs of size $n$, and proper minor closed classes. In this talk, we look at developing a structural understanding of twin-width in terms of induced subdivisions.
Structural characterizations of graph parameters have mostly looked at graph minors, for instance, bounded tree-width graphs are exactly those forbidding a large wall minor. An analogue in terms of induced subgraphs could be that, for sparse graphs, large treewidth implies the existence of an induced subdivision of a large wall. However, Sintiari and Trotignon have ruled out such a characterization by showing the existence of graphs with arbitrarily large girth avoiding any induced subdivision of a theta ($K_{2,3}$). Abrishami, Chudnovsky, Hajebi and Spirkl have recently shown that such (theta, triangle)-free classes have nevertheless logarithmic treewidth.
After an introduction to twin-width and its ties to vertex orderings, we show that theta-free graphs of girth at least 5 have bounded twin-width.
Joint work with Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé and Rémi Watrigant.

Chengfei Xie, On the packing densities of superballs in high dimensions
2022-06-22 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
The sphere packing problem asks for the densest packing of nonoverlapping equal-sized balls in the space. This is an old and difficult problem in discrete geometry. In this talk, we give a new proof for the result that for $ 1<p<2 $, the translative packing density of superballs (a generalization of $ell^p$ balls) in $mathbb{R}^n$ is $Omega(n/2^n)$.
This is joint work with Gennian Ge.

Ben Lund, Radial projections in finite space
2022-06-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss several results on the general question: For how many points $y$ can $|pi_y(E)|$ be much smaller than this maximum?
This is motivated by an analogous question in fractal geometry. The Hausdorff dimension of a radial projection of a set $E$ in $n$ dimensional real space will typically be the minimum of $n-1$ and the Hausdorff dimension of $E$. Several recent papers by authors including Matilla, Orponen, Liu, Shmerikin, and Wang consider the question: How large can the set of points with small radial projections be? This body of work has several important applications, including recent progress on the Falconer distance conjecture.
This is joint with Thang Pham and Vu Thi Huong Thu.

Xizhi Liu, Hypergraph Turán problem: from 1 to ∞
2022-06-29 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
One interesting difference between (nondegenerate) Graph Turán problem and Hypergraph Turán problem is that the hypergraph families can have at least two very different extremal constructions. In this talk, we will look at some recent progress and approaches to constructing hypergraph families with at least two different extremal constructions.
Based on some joint work with Dhruv Mubayi, Christian Reiher, Jianfeng Hou, Heng Li, and Yixiao Zhang.

Eric Vigoda, Computational phase transition and MCMC algorithms
2022-07-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
This talk will highlight recent results establishing a beautiful computational phase transition for approximate counting/sampling in (binary) undirected graphical models (such as the Ising model or on weighted independent sets). The computational problem is to sample from the equilibrium distribution of the model or equivalently approximate the corresponding normalizing factor known as the partition function. We show that when correlations die off on the infinite D-regular tree then the Gibbs sampler has optimal mixing time of $O(n log n)$ on any graph of maximum degree D, whereas when the correlations persist (in the limit) then the sampling/counting problem are NP-hard to approximate. The Gibbs sampler is a simple Markov Chain Monte Carlo (MCMC) algorithm. Key to these mixing results are a new technique known as spectral independence which considers the pairwise influence of vertices. We show that spectral independence implies an optimal convergence rate for a variety of MCMC algorithms.

Sepehr Hajebi, Holes, hubs and bounded treewidth
2022-07-07 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
A hole in a graph $G$ is an induced cycle of length at least four, and for every hole $H$ in $G$, a vertex $hin Gsetminus H$ is called a $t$-hub for $H$ if $h$ has at least $t$ neighbor in $H$. Sintiari and Trotignon were the first to construct graphs with arbitrarily large treewidth and no induced subgraph isomorphic to the “basic obstructions,” that is, a fixed complete graph, a fixed complete bipartite graph (with parts of equal size), all subdivisions of a fixed wall and line graphs of all subdivisions of a fixed wall. They named their counterexamples “layered wheels” for a good reason: layered wheels contain wheels in abundance, where a wheel means a hole with a $3$-hub. In accordance, one may ask whether graphs with no wheel and no induced subgraph isomorphic to the basic obstructions have bounded treewidth. This was also disproved by a recent construction due to Davies. But holes with a $2$-hub cannot be avoided in graphs with large treewidth: graphs containing no hole with a $2$-hub and no induced subgraph isomorphic to the basic obstructions have bounded treewidth. I will present a proof of this result, and will also give an overview of related works.
Based on joint work with Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sophie Spirkl and Kristina Vušković.

Kevin Hendrey, Product Structure of Graph Classes with Bounded Treewidth
2022-07-11 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The strong product $Gboxtimes H$ of graphs $G$ and $H$ is the graph on the cartesian product $V(G)times V(H)$ such that vertices $(v,w)$ and $(x,y)$ are adjacent if and only if $max{d_G(v,x),d_H(w,y)}=1$. Graph product structure theory aims to describe complicated graphs in terms of subgraphs of strong products of simpler graphs. This area of research was initiated by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood, who showed that every planar graph is a subgraph of the strong product of a $Hboxtimes Pboxtimes K_3$ for some path $P$ and some graph $H$ of treewidth at most $3$. In this talk, I will discuss the product structure of various graph classes of bounded treewidth. As an example, we show that there is a function $f:mathbb{N}rightarrow mathbb{N}$ such that every planar graph of treewidth at most $k$ is a subgraph of $Hboxtimes K_{f(k)}$ for some graph $H$ of treewidth at most $3$.
This is based on joint work with Campbell, Clinch, Distel, Gollin, Hickingbotham, Huynh, Illingworth, Tamitegama, Tan and Wood.

Jinyoung Park (박진영), Thresholds 1/2
2022-07-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its “expectation-threshold,” which is a natural (and often easy to calculate) lower bound on the threshold.
In the first talk on Monday, I will introduce the Kahn-Kalai Conjecture with some motivating examples and then briefly talk about the recent resolution of the Kahn-Kalai Conjecture due to Huy Pham and myself.
In the second talk on Tuesday, I will discuss our proof of the conjecture in detail.

Jinyoung Park (박진영), Thresholds 2/2
2022-07-19 14:00 - 16:00 | Room B332, IBS (기초과학연구원)

Noam Lifshitz, Product free sets in the alternating group
2022-07-27 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of $A_n$ be?
In the talk we will completely solve the problem by determining the largest product free subset of $A_n$.
Our proof combines a representation theoretic argument due to Gowers, with an analytic tool called hypercontractivity for global functions. We also make use of a dichotomy between structure and a pseudorandomness notion of functions over the symmetric group known as globalness.
Based on a joint work with Peter Keevash and Dor Minzer.

Seunghun Lee (이승훈), Inscribable order types
2022-08-01 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this talk, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $Theta(frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability. This is a joint work with Michael Gene Dobbins.

Lars Jaffke, Taming graphs with no large creatures and skinny ladders
2022-08-03 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class $mathcal{G}$ there exists a constant $k$ such that no member of $mathcal{G}$ contains a $k$-creature as an induced subgraph or a $k$-skinny-ladder as an induced minor, then there exists a polynomial $p$ such that every $G in mathcal{G}$ contains at most $p(|V(G)|)$ minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from $mathcal{G}$. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).
Joint work with Jakub Gajarský, Paloma T. Lima, Jana Novotná, Marcin Pilipczuk, Paweł Rzążewski, and Uéverton S. Souza.

Eun Jung Kim (김은정), Directed flow-augmentation
2022-08-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that, given a directed graph G, two integers $s,tin V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$, with probability $2^{−operatorname{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph.
The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems:
(1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA’13, Algorithmica’17],
(2) a number of weighted variants of classic directed cut problems, such as Weighted st-Cut, Weighted Directed Feedback Vertex Set, or Weighted Almost 2-SAT.
By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H, if the List H-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable.
Joint work with Stefan Kratsch, Marcin Pilipczuk, Magnus Wahlström.
