Discrete Mathematics Group

Hugo Jacob, On the parameterized complexity of computing tree-partitions
2022-11-09 16:30 - 17:30 (Asia/Seoul) | Zoom ID: 869 4632 6610 (ibsdimag)
Following some recent FPT algorithms parameterized by the width of a given tree-partition due to Bodlaender, Cornelissen, and van der Wegen, we consider the parameterized problem of computing a decomposition. We prove that computing an optimal tree-partition is XALP-complete, which is likely to exclude FPT algorithms. However, we prove that computing a tree-partition of approximate width is tractable using a relatively simple sketch. This is sufficient to remove the requirement of having a given tree-partition for FPT algorithms. Our simple sketch can be adapted for several regimes within polynomial time and FPT time. Furthermore, we adapt some simple structural results about the tree-partition-width of subdivisions, and use them to compare tree-cut width and tree-partition-width.
Based on joint work with Hans Bodlaender and Carla Groenland.

Jungho Ahn (안정호), Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes
2022-11-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let $mathcal{F}$ be a family of graphs, and let $p$ and $r$ be nonnegative integers.
The $(p,r,mathcal{F})$-Covering problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^psetminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $mathcal{F}$, where $G^p$ is the $p$-th power of $G$ and $N^r_G[D]$ is the set of all vertices in $G$ at distance at most $r$ from $D$ in $G$. The $(p,r,mathcal{F})$-Packing problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $mathcal{F}$, and for distinct $i,jin {1, ldots, k}$, the distance between $V(H_i)$ and $V(H_j)$ in $G$ is larger than $r$. The $(p,r,mathcal{F})$-Covering problem generalizes Distance-$r$ Dominating Set and Distance-$r$ Vertex Cover, and the $(p,r,mathcal{F})$-Packing problem generalizes Distance-$r$ Independent Set and Distance-$r$ Matching. By taking $(p’,r’,mathcal{F}’)=(pt, rt, mathcal{F})$, we may formulate the $(p,r,mathcal{F})$-Covering and $(p, r, mathcal{F})$-Packing problems on the $t$-th power of a graph. Moreover, $(1,0,mathcal{F})$-Covering is the $mathcal{F}$-Free Vertex Deletion problem, and $(1,0,mathcal{F})$-Packing is the Induced-$mathcal{F}$-Packing problem.
We show that for every fixed nonnegative integers $p,r$ and every fixed nonempty finite family $mathcal{F}$ of connected graphs, the $(p,r,mathcal{F})$-Covering problem with $pleq2r+1$ and the $(p,r,mathcal{F})$-Packing problem with $pleq2lfloor r/2rfloor+1$ admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size $k$. We obtain the same kernels for their annotated variants. As corollaries, we prove that Distance-$r$ Vertex Cover, Distance-$r$ Matching, $mathcal{F}$-Free Vertex Deletion, and Induced-$mathcal{F}$-Packing for any fixed finite family $mathcal{F}$ of connected graphs admit almost linear kernels on every nowhere dense class of graphs and linear kernels on every class of graphs with bounded expansion. Our results extend the results for Distance-$r$ Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for Distance-$r$ Independent Set by Pilipczuk and Siebertz (EJC 2021).
This is joint work with Jinha Kim and O-joung Kwon.

Dabeen Lee (이다빈), Non-smooth and Hölder-smooth submodular optimization
2022-10-27 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
We study the problem of maximizing a continuous DR-submodular function that is not necessarily smooth. We prove that the continuous greedy algorithm achieves an [(1−1/e)OPT−ϵ] guarantee when the function is monotone and Hölder-smooth, meaning that it admits a Hölder-continuous gradient. For functions that are non-differentiable or non-smooth, we propose a variant of the mirror-prox algorithm that attains an [(1/2)OPT−ϵ] guarantee. We apply our algorithmic frameworks to robust submodular maximization and distributionally robust submodular maximization under Wasserstein ambiguity. In particular, the mirror-prox method applies to robust submodular maximization to obtain a single feasible solution whose value is at least (1/2)OPT−ϵ. For distributionally robust maximization under Wasserstein ambiguity, we deduce and work over a submodular-convex maximin reformulation whose objective function is Hölder-smooth, for which we may apply both the continuous greedy method and the mirror-prox method.
Joint work with Duksang Lee and Nam Ho-Ngyuen.

Florent Koechlin, Uniform random expressions lack expressivity
2022-10-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In computer science, random expressions are commonly used to analyze algorithms, either to study their average complexity, or to generate benchmarks to test them experimentally. In general, these approaches only consider the expressions as purely syntactic trees, and completely ignore their semantics — i.e. the mathematical object represented by the expression.
However, two different expressions can be equivalent (for example “0*(x+y)” and “0” represent the same expression, the null expression). Can these redundancies question the relevance of the analyses and tests that do not take into account the semantics of the expressions?
I will present how the uniform distribution over syntactic expression becomes completely degenerate when we start taking into account their semantics, in a very simple but common case where there is an absorbing element. If time permits it, I will briefly explain why the BST distribution offers more hope.
This is a joint work with Cyril Nicaud and Pablo Rotondo.

Xavier Goaoc, Order types and their symmetries
2022-10-13 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
Order types are a combinatorial classification of finite point sets used in discrete and computational geometry. This talk will give an introduction to these objects and their analogue for the projective plane, with an emphasis on their symmetry groups.
This is joint work with Emo Welzl.

Nika Salia, Exact results for generalized extremal problems forbidding an even cycle
2022-10-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s ge 2$ and sufficiently large $n$. Moreover, for $sin{2,3}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph.
This is joint work with Ervin Győri (Renyi Institute), Zhen He (Tsinghua University), Zequn Lv (Tsinghua University), Casey Tompkins (Renyi Institute), Kitti Varga (Technical University of Budapest BME), and Xiutao Zhu (Nanjing University).

Konstantin Tikhomirov, A remark on the Ramsey number of the hypercube
2022-10-06 10:00 - 11:00 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
A well-known conjecture of Burr and Erdős asserts that the Ramsey number $r(Q_n)$ of the hypercube $Q_n$ on $2^n$ vertices is of the order $O(2^n)$. In this paper, we show that $r(Q_n)=O(2^{2n−cn})$ for a universal constant $c>0$, improving upon the previous best-known bound $r(Q_n)=O(2^{2n})$, due to Conlon, Fox, and Sudakov.

Zixiang Xu (徐子翔), On the degenerate Turán problems
2022-10-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a graph $F$, the Turán number is the maximum number of edges in an $n$-vertex simple graph not containing $F$. The celebrated Erdős-Stone-Simonovits Theorem gives that [ text{ex}(n,F)=bigg(1-frac{1}{chi(F)-1}+o(1)bigg)binom{n}{2},] where $chi(F)$ is the chromatic number of $H$. This theorem asymptotically solves the problem when $chi(F)geqslant 3$. In case of bipartite graphs $F$, not even the order of magnitude is known in general. In this talk, I will introduce some recent progress on Turán numbers of bipartite graphs and related generalizations and discuss several methods developed in recent years. Finally, I will introduce some interesting open problems on this topic.

Santiago Guzmán-Pro, Local expressions of graphs classes
2022-09-29 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
A common technique to characterize hereditary graph classes is to exhibit their minimal obstructions. Sometimes, the set of minimal obstructions might be infinite, or too complicated to describe. For instance, for any $kge 3$, the set of minimal obstructions of the class of $k$-colourable graphs is yet unknown. Nonetheless, the Roy-Gallai-Hasse-Vitaver Theorem asserts that a graph $G$ is $k$-colourable if and only if it admits an orientation with no directed walk on $k+1$ vertices. We say that a class of graphs $mathcal{P}$ is expressible by forbidden orientations if there is a finite set $F$ of oriented graphs such that $mathcal{P}$ is the class of graphs that admit an $F$-free orientation. We are interested in understanding which graph classes are expressible by forbidden orientations (and why). In this talk, we present some limitations of this expression system. In particular, we show that the class of even-hole free graphs is not expressible by forbidden orientations.
Throughout the talk, we also mention some other related expression systems. In particular, each of these systems provides a certification method to the $mathcal{P}$-decision problem; i.e., decide if an input graph belongs to the class $mathcal{P}$. We conclude this talk by proposing a general framework to talk about these expression systems. This framework allows us to formalize the question, what can be certified this way?
Based on a joint work with César Hernández-Cruz.

Alexander Clifton, Ramsey Theory for Diffsequences
2022-09-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Van der Waerden’s theorem states that any coloring of $mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r,k)$ which is the smallest $n$ such that any $r$-coloring of ${1,2,cdots,n}$ guarantees the presence of a monochromatic arithmetic progression of length $k$.
It is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion, introduced by Landman and Robertson, is that of a $D$-diffsequence, which is an increasing sequence $a_1<a_2<cdots<a_k$ in which the consecutive differences $a_i-a_{i-1}$ all lie in some given set $D$. We say that $D$ is $r$-accessible if every $r$-coloring of $mathbb{N}$ contains arbitrarily long monochromatic $D$-diffsequences. When $D$ is $r$-accessible, we define $Delta(D,k;r)$ as the smallest $n$ such that any $r$-coloring of ${1,2,cdots,n}$ guarantees the presence of a monochromatic $D$-diffsequence of length $k$.
One question of interest is to determine the possible behaviors of $Delta$ as a function of $k$. In this talk, we will demonstrate that is possible for $Delta(D,k;r)$ to grow faster than polynomial in $k$. We will also discuss a broad class of $D$’s which are not $2$-accessible.

Mehtaab Sawhney, Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture
2022-09-21 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $Clog_2 n$ (i.e., if it has near-optimal Ramsey behavior). We study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a $C$-Ramsey graph. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős offered one of his notorious monetary prizes and the proof involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation.
Joint w. Matthew Kwan, Ashwin Sah, and Lisa Sauermann

Sebastian Wiederrecht, Killing a vortex
2022-09-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every $tinmathbb{N},$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most $c_{t}$ vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most $c_{t}$ and “at most $c_{t}$ vortices of depth $c_{t}$”. Our main combinatorial result is a “vortex-free” refinement of the above structural theorem as follows: we identify a (parameterized) graph $H_{t}$, called shallow vortex grid, and we prove that if in the above structural theorem we replace $K_{t}$ by $H_{t},$ then the resulting decomposition becomes “vortex-free”. Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some $H_{t},$ the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an $H_{t}$-minor-free graph $G$, computes the generating function of all perfect matchings of $G$ in polynomial time. This algorithm yields, on $H_{t}$-minor-free graphs, polynomial algorithms for computational problems such as the {dimer problem, the exact matching problem}, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $H_{t}$ as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.
This is joint work with Dimitrios M. Thilikos.

Dömötör Pálvölgyi, C-P3O: Orientation of convex sets and other good covers
2022-09-07 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a natural interiority condition. Such systems, P3O (partial 3-order), are a natural generalization of posets, and include the order types of planar point sets. Our main result is that P3O that emerge from points sets, p-P3O, and P3O that emerge from convex sets, C-P3O, do not contain each other. We also extend our orientation to other good covers from convex sets and study the resulting P3O’s.
Based on joint work with Agoston, Damasdi, and Keszegh:
https://arxiv.org/abs/2206.01721
https://arxiv.org/abs/2206.01723

Bjarne Schülke, A local version of Katona's intersection theorem
2022-09-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Katona’s intersection theorem states that every intersecting family $mathcal Fsubseteq[n]^{(k)}$ satisfies $vertpartialmathcal Fvertgeqvertmathcal Fvert$, where $partialmathcal F={Fsetminus x:xin Finmathcal F}$ is the shadow of $mathcal F$.
Frankl conjectured that for $n>2k$ and every intersecting family $mathcal Fsubseteq [n]^{(k)}$, there is some $iin[n]$ such that $vert partial mathcal F(i)vertgeq vertmathcal F(i)vert$, where $mathcal F(i)={Fsetminus i:iin Finmathcal F}$ is the link of $mathcal F$ at $i$.
Here, we prove this conjecture in a very strong form for $n> binom{k+1}{2}$.
In particular, our result implies that for any $jin[k]$, there is a $j$-set ${a_1,dots,a_j}in[n]^{(j)}$ such that [ vert partial mathcal F(a_1,dots,a_j)vertgeq vertmathcal F(a_1,dots,a_j)vert.]A similar statement is also obtained for cross-intersecting families.

Raphael Steiner, Congruence-constrained subdivisions in digraphs
2022-08-31 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
I will present the short proof from [1] that for every digraph F and every assignment of pairs of integers $(r_e,q_e)_{ein A(F)}$ to its arcs, there exists an integer $N$ such that every digraph D with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$ for every $e in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs and at the same time yields a novel proof of his result. I will also talk about how a hypergraph coloring result from [2] may help to obtain good bounds on $N$ in the special case when $F$ is subcubic.
[1] https://arxiv.org/abs/2208.06358
[2] https://arxiv.org/abs/2206.13635

Jun Gao, Number of (k-1)-cliques in k-critical graph
2022-08-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove that for $n>kgeq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.
This is joint work with Jie Ma.

Brett Leroux, Expansion of random 0/1 polytopes
2022-08-25 10:00 - 11:00 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
A conjecture of Milena Mihail and Umesh Vazirani states that the edge expansion of the graph of every $0/1$ polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a $0/1$ polytope in $mathbb{R}^d$ is greater than 1 over some polynomial function of $d$. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random $0/1$ polytope in $mathbb{R}^d$ is at least $frac{1}{12d}$ with high probability.
After discussing this result and the proof, we will mention some possible extensions. To conclude, we will discuss some related questions about the combinatorics of random polytopes, including the diameter problem.
This is joint work with Luis Rademacher.

Raul Lopes, Temporal Menger and related problems
2022-08-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A temporal graph is a graph whose edges are available only at specific times. In this scenario, the only valid walks are the ones traversing adjacent edges respecting their availability, i.e. sequence of adjacent edges whose appearing times are non-decreasing.
Given a graph G and vertices s and t of G, Menger’s Theorem states that the maximum number of (internally) vertex disjoint s,t-paths is equal to the minimum size of a subset X for which G-X contains no s,t-path. This is a classical result in Graph Theory, taught in most basic Graph Theory courses, and it holds also when G is directed and when edge disjoint paths and edge cuts are considered instead. A direct translation of Menger’s Theorem to the temporal context has been known not to hold since an example was shown in the seminal paper by Kempe, Kleinberg and Kumar (STOC’00). In this talk, an overview of possible temporal versions of Menger’s Theorem will be discussed, as well as the complexity of the related problems.

Noleen Köhler, Testing first-order definable properties on bounded degree graphs
2022-08-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Property testers are probabilistic algorithms aiming to solve a decision problem efficiently in the context of big-data. A property tester for a property P has to decide (with high probability correctly) whether a given input graph has property P or is far from having property P while having local access to the graph. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree model. We show that any FO property that is defined by a formula with quantifier prefix ∃*∀* is testable, while there exists an FO property that is expressible by a formula with quantifier prefix ∀*∃* that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs.
This is joint work with Isolde Adler and Pan Peng.

Akash Kumar, Random walks and Forbidden Minors
2022-08-10 16:30 - 17:30 | Zoom ID: 870 0312 9412 (ibsecopro) [CLOSED]
Random walks and spectral methods have had a strong influence on modern graph algorithms as evidenced by the extensive literature on the subject. In this talk, I will present how random walks helped make progress on algorithmic problems on planar graphs.
In particular, I show how random walk based (i.e., spectral) approaches led to progress on finding forbidden minors [K.-Seshadhri-Stolman, FOCS 2018] as well as on deciding planarity [K.-Seshadhri-Stolman, STOC 2019] in bounded degree graphs within the property testing framework. I will also cover how these approaches eventually led to progress on the so-called “efficient partition oracle” problem [K.-Seshadhri-Stolman, FOCS 2021].
The talk will assume minimal background by presenting a stand-alone story that should be of interest to students/researchers in computer science.

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.

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.

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.

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.

Jinyoung Park (박진영), Thresholds 2/2
2022-07-19 14:00 - 16:00 | 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 1/2
2022-07-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)

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.

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.

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.

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.
