Discrete Mathematics Group

Tuukka Korhonen, Fast FPT-Approximation of Branchwidth
2021-11-25 16:30 - 17:30 (Asia/Seoul) | Zoom ID: 869 4632 6610 (ibsdimag)
Branchwidth determines how graphs, and more generally, arbitrary connectivity (basically symmetric and submodular) functions could be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable (FPT) 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations could decrease the width of a branch decomposition or that the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework.

An algorithm that for a given n-vertex graph G and integer k in time $2^{2^{O(k)}} n^2$ either constructs a rank decomposition of G of width at most 2k or concludes that the rankwidth of G is more than $k$. It also yields a $(2^{2k+1}−1)$-approximation algorithm for cliquewidth within the same time complexity, which in turn, improves to $f(k) n^2$ the running times of various algorithms on graphs of cliquewidth $k$. Breaking the “cubic barrier” for rankwidth and cliquewidth was an open problem in the area.
An algorithm that for a given n-vertex graph G and integer k in time $2^{O(k)} n$ either constructs a branch decomposition of G of width at most $2k$ or concludes that the branchwidth of G is more than $k$. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].

This is joint work with Fedor Fomin.

Casey Tompkins, Ramsey numbers of Boolean lattices
2021-11-23 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The poset Ramsey number $R(Q_{m},Q_{n})$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_{N}$ has a blue induced copy of $Q_{m}$ or a red induced copy of $Q_{n}$. Axenovich and Walzer showed that $n+2le R(Q_{2},Q_{n})le2n+2$. Recently, Lu and Thompson
improved the upper bound to $frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_{2},Q_{n})=n+O(n/log n)$.
Joint work with Dániel Grósz and Abhishek Methuku.

Graph Product Structure Theory: Gathering of Participants from Korea
2021-11-23 - 2021-11-26 | Room B223, IBS (기초과학연구원)
On November 22-26, 2021, there is a “Graph Product Structure Theory” workshop in BIRS Centre in Banff (https://www.birs.ca/events/2021/5-day-workshops/21w5235), organized in a hybrid manner with 15 onsite participants and around 50 remote participants. We would like to meet in a group of 5-10 remote participants from Korea in one place, creating a secondary workshop site in Korea. This would allow joint participation in online talks and in-person discussions between local participants.

Sebastian Wiederrecht, Matching Minors in Bipartite Graphs
2021-11-11 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
Matching minors are a specialisation of minors which preserves the existence and elementary structural properties of perfect matchings. They were first discovered as part of the study of the Pfaffian recognition problem on bipartite graphs (Polya’s Permanent Problem) and acted as a major inspiration for the definition of butterfly minors in digraphs. In this talk we consider the origin and motivation behind the study of matching minors, the current state of the art, and their relation to structural digraph theory. The main result is a generalisation of the structure theorem by Robertson et al. and McCuaig for $K_{3,3}$-matching minor free bipartite graphs to bipartite graphs excluding $K_{t,t}$ as a matching minor for general t. This generalisation can be seen as a matching theoretic version of the Flat Wall Theorem by Robertson and Seymour.

Jaehoon Kim (김재훈), 2-complexes with unique embeddings in 3-space
2021-11-09 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem. This is joint work with Agelos Georgakopoulos.

Martin Milanič, Tree Decompositions with Bounded Independence Number
2021-11-05 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The independence number of a tree decomposition $mathcal{T}$ of a graph is the smallest integer $k$ such that each bag of $mathcal{T}$ induces a subgraph with independence number at most $k$. If a graph $G$ is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Set (MWIS) problem can be solved in polynomial time. Motivated by this observation, we consider six graph containment relations—the subgraph, topological minor, and minor relations, as well as their induced variants—and for each of them characterize the graphs $H$ for which any graph excluding $H$ with respect to the relation admits a tree decomposition with bounded independence number. Furthermore, using a variety of tools including SPQR trees and potential maximal cliques, we show how to obtain such tree decompositions efficiently.
As an immediate consequence, we obtain that the MWIS problem can be solved in polynomial time in an infinite family of graph classes that properly contain the class of chordal graphs. In fact, our approach shows that the Maximum Weight Independent $mathcal{H}$-Packing problem, a common generalization of the MWIS and the Maximum Weight Induced Matching problems, can be solved in polynomial time in these graph classes.
This is joint work with Clément Dallard and Kenny Štorgel.

Ben Lund, Maximal 3-wise intersecting families
2021-11-02 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A family $mathcal F$ of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from $mathcal F$ has a common element, and moreover, no set can be added to $mathcal F$ while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a maximal k-wise intersecting family, for k≥3. We resolve this problem for k=3 and n even and sufficiently large.
This is joint work with Kevin Hendrey, Casey Tompkins, and Tuan Tran.

Donggyu Kim (김동규), 𝝘-graphic delta-matroids and their applications
2021-10-26 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Bouchet (1987) defined delta-matroids by relaxing the base exchange axiom of matroids. Oum (2009) introduced a graphic delta-matroid from a pair of a graph and its vertex subset. We define a $Gamma$-graphic delta-matroid for an abelian group $Gamma$, which generalizes a graphic delta-matroid.
For an abelian group $Gamma$, a $Gamma$-labelled graph is a graph whose vertices are labelled by elements of $Gamma$. We prove that a certain collection of edge sets of a $Gamma$-labelled graph forms a delta-matroid, which we call a $Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet (1987) to find a maximum weight feasible set in such a delta-matroid. We also prove that a $Gamma$-graphic delta-matroid is a graphic delta-matroid if and only if it is even. We prove that every $mathbb{Z}_p^k$-graphic delta matroid is represented by some symmetric matrix over a field of characteristic of order $p^k$, and if every $Gamma$-graphic delta-matroid is representable over a finite field $mathbb{F}$, then $Gamma$ is isomorphic to $mathbb{Z}_p^k$ and $mathbb{F}$ is a field of order $p^ell$ for some prime $p$ and positive integers $k$ and $ell$.
This is joint work with Duksang Lee and Sang-il Oum.

Young Researchers in Extremal and Probabilistic Combinatorics
2021-10-20 - 2021-10-22 | Zoom ID: 554 788 7710 (507464)
The aim of the Young Researchers in Extremal and Probabilistic Combinatorics is to bring together early career researchers working on these topics. The workshop will consist of several 25 minute talks across three days from October 20 to 22, 2021. Due to Covid the workshop will be held online.

Invited Speakers & Program
Oct. 20 Wednesday 4 PM (KST) – 8 PM (KST)

Andrzej Grzesik (Jagiellonian University): Degenerated generalized Turán numbers of cycles, 4:00-4:25
Jan Volec (Czech Technical University): Existence of common graphs with large chromatic number, 4:30-4:55
Balázs Patkós (Rényi Institute): Vector sum-intersection theorems, 5:00-5:25
István Tomon (ETH Zurich): Small doubling, atomic structure and $ell$-divisible set families, 5:30-5:55
Michael Anastos (Freie Universität Berlin): Longest Cycles in Sparse Random Graphs and Where to Find Them, 6:30-6:55
Adam Zsolt Wagner (Tel Aviv University): Constructions in combinatorics via neural networks, 7:00-7:25
Yelena Yuditsky (Université libre de Bruxelles): On multicolor Ramsey numbers and subset-coloring of hypergraphs, 7:30-7:55

Oct. 21 Thursday 4 PM (KST) – 8 PM (KST)

Kevin Hendrey (IBS Discrete Mathematics Group): Extremal functions for sparse minors, 4:00-4:25
Simona Boyadzhiyska (Freie Universität Berlin): Ramsey simplicity of random graphs, 4:30-4:55
Shagnik Das (National Taiwan University): Schur’s Theorem in randomly perturbed sets, 5:00-5:25
Tony Huynh (Monash University): Subgraph densities in minor-closed classes (and beyond), 5:30-5:55
Ander Lamaison (Masaryk University): Hypergraphs with minimum uniform Turán density, 6:30-6:55
Ben Lund (IBS Discrete Mathematics Group): Maximal 3-wise intersecting families, 7:00-7:25
Liana Yepremyan (London School of Economics): Enumerating independent sets in Abelian Cayley graphs, 7:30-7:55

Oct. 22 Friday 4 PM (KST) – 8 PM (KST)

Andrey Kupavskii (CNRS): Binary scalar products, 4:00-4:25
Dániel Gerbner (Rényi Institute): Exact results for generalized Turán problems, 4:30-4:55
Oliver Janzer (University of Cambridge): Tiling with monochromatic bipartite graphs of bounded maximum degree, 5:00-5:25
Nika Salia (Rényi Institute): Pósa-type results for Berge Hypergraphs, 5:30-5:55
Debsoumya Chakraborti (IBS Discrete Mathematics Group): Mixing time and expanders, 6:30-6:55
Wei-Tian Li (National Chung Hsing University): Ramsey Properties for $V$-shaped Posets in the Boolean Lattices, 7:00-7:25
Dimitry Zakharov (Moscow Institute of Physics and Technology): Zero subsums in vector spaces over finite fields, 7:30-7:55

Abstracts (pdf file)

Joonkyung Lee (이준경), Majority dynamics on sparse random graphs
2021-10-12 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph $G(n,p)$, the random initial $pm 1$-assignment converges to a $99%$-agreement with high probability whenever $p=omega(1/n)$.
This conjecture was first confirmed for $pgeqlambda n^{-1/2}$ for a large constant $lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< lambda n^{-1/2}$. We break this $Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $lambda’ n^{-3/5}log n leq p leq lambda n^{-1/2}$ with a large constant $lambda’>0$.
Joint work with Debsoumya Chakraborti, Jeong Han Kim and Tuan Tran.

Paul Seymour, Polynomial bounds for chromatic number
2021-10-08 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
The Gyárfás-Sumner conjecture says that for every forest $H$, there is a function $f$ such that the chromatic number $chi(G)$ is at most $f(omega(G))$ for every $H$-free graph $G$ (“$H$-free” means with no induced subgraph isomorphic to $H$, and $omega(G)$ is the size of the largest clique of $G$). This well-known conjecture has been proved only for a few types of forest.
Nevertheless, there is a much stronger conjecture, due to Esperet: that for every forest $H$, there is a polynomial function $f$ as above. As one might expect, this has been proved for even fewer types of forest; and the smallest tree $H$ for which Esperet’s conjecture is not known is the five-vertex path $P_5$.
A third notorious conjecture is the Erdős-Hajnal conjecture, that for every graph $H$, there exists $c>0$ such that $alpha(G)omega(G)ge |G|^c$ for every $H$-free graph $G$ (where $alpha(G)$ is the size of the largest stable set of $G$). The smallest graph $H$ for which this is not known is also $P_5$, which, conveniently, is a forest; and every forest that satisfies Esperet’s conjecture also satisfies the Erdős-Hajnal conjecture. So there is substantial interest in the chromatic numbers of $P_5$-free graphs. The best upper bound that was previously known, due to Esperet, Lemoine, Maffray, and Morel, was that $chi(G)le (5/27)3^omega(G)$ for every $P_5$-free graph $G$ with $omega(G) > 2$. In recent work with Alex Scott and Sophie Spirkl, we have proved several results related to Esperet’s conjecture, including proofs of its truth for some new types of forest $H$, and a “near-polynomial” bound when $H = P_5$, that $chi(G) le omega(G)^{log_2(omega(G))}$ for every $P_5$-free graph $G$ with $omega(G) > 2$. We survey these results and give a proof of the new bound for $P_5$.

Eunjin Oh (오은진), Feedback Vertex Set on Geometric Intersection Graphs
2021-10-05 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(sqrt{k}log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].

Péter Pál Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities
2021-09-30 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The Alon-Jaeger-Tarsi conjecture states that for any finite field $mathbb{F}$ of size at least 4 and any nonsingular matrix $M$ over $mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Mx$ has a 0 component. In joint work with János Nagy we proved this conjecture when the size of the field is sufficiently large, namely, when $61

Kevin Hendrey, Extremal functions for sparse minors
2021-09-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, [c(H) = (1+o(1))max (v(H)/2, v(H)-alpha(H)),] extending recent results of Haslegrave, Kim and Liu (2020). Joint work with Sergey Norin and David R. Wood.

Dabeen Lee (이다빈), Mixing sets, submodularity, and chance-constrained optimization
2021-09-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this talk, we first explain basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull. This is based on joint work with Fatma Fatma Kılınç-Karzan and Simge Küçükyavuz.

Cheolwon Heo (허철원), Representations of even-cycle matroids
2021-08-31 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A signed graph is a pair $(G,Sigma)$ where $G$ is a graph and $Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,Sigma)$ if $|C cap Sigma|$ is even; otherwise, $C$ is odd. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,Sigma)$. For even-cycle matroids, two fundamental questions arise:
(1) what is the relationship between two signed graphs representing the same even-cycle matroids?
(2) how many signed graphs can an even-cycle matroid have?
For (a), we characterize two signed graphs $(G_1,Sigma_1)$ and $(G_2,Sigma_2)$ where $G_1$ and $G_2$ are $4$-connected that represent the same even-cycle matroids.
For (b), we introduce pinch-graphic matroids, which can generate exponentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a pair of vertices such that every odd cycle intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. This is joint work with Bertrand Guenin and Irene Pivotto.

Eun Jung Kim (김은정), A Constant-factor Approximation for Weighted Bond Cover
2021-08-24 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The Weighted $mathcal F$-Vertex Deletion for a class $mathcal F$ of graphs asks, given a weighted graph $G$, for a minimum weight vertex set $S$ such that $G-Sinmathcal F$. The case when $mathcal F$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $mathcal F$-Vertex Deletion. Only three cases of minor-closed $mathcal F$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set.
We study the problem for the class $mathcal F$ of $theta_c$-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA’14] which states the following: any graph $G$ containing a $theta_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $theta_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $mathcal F$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
This is joint work with Euiwoong Lee and Dimitrios M. Thilikos.

Petr Hliněný, Twin-width is linear in the poset width
2021-08-18 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced quite recently, in 2020 by Bonnet, Kim, Thomassé, and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing a prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose double-exponential bound.
We prove that posets of width d have twin-width at most 9d with a direct and elementary argument, and show that this bound is tight up to a constant factor. Specifically, for posets of width 2, we prove that in the worst case their twin-width is also equal to 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset.
(Joint work with my student Jakub Balaban who obtained the main ideas in his bachelor thesis.)

Linda Cook, Two results on graphs with holes of restricted lengths
2021-08-17 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc provided a structural description of the class of even-hole-free graphs. I will describe the structure of all graphs that contain only holes of length $ell$ for every $ell geq 7$ (joint work with Jake Horsfield, Myriam Preissmann, Paul Seymour, Ni Luh Dewi Sintiari, Cléophée Robin, Nicolas Trotignon, and Kristina Vuškovíc.
Analysis of how holes interact with graph structure has yielded detection algorithms for holes of various lengths and parities. In 1991, Bienstock showed it is NP-Hard to test whether a graph G has an even (or odd) hole containing a specified vertex $v in V(G)$. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc gave a polynomial-time algorithm to recognize even-hole-free graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi, and Seymour provided a simpler and slightly faster algorithm to test whether a graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour, and Spirkl provided a polynomial-time algorithm to test whether a graph contains an odd hole. Later that year, Chudnovsky, Scott, and Seymour strengthened this result by providing a polynomial-time algorithm to test whether a graph contains an odd hole of length at least $ell$ for any fixed integer $ell geq 5$. I will present a polynomial-time algorithm (joint work with Paul Seymour) to test whether a graph contains an even hole of length at least $ell$ for any fixed integer $ell geq 4$.

Duksang Lee (이덕상), Intertwining connectivities for vertex-minors and pivot-minors
2021-08-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.

Maria Chudnovsky, Induced subgraphs and tree decompositions
2021-07-28 15:00 - 16:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Tree decompositions are a powerful tool in structural graph theory; they are traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has until recently remained out of reach.
Tree decompositions are closely related to the existence of “laminar collections of separations” in a graph, which roughly means that the separations in the collection “cooperate” with each other, and the pieces that are obtained when the graph is simultaneously decomposed by all the separations in the collection “line up” to form a tree structure. Such collections of separations come up naturally in the context of forbidden minors.
In the case of families where induced subgraphs are excluded, while there are often natural separations, they are usually very far from forming a laminar collection. In what follows we mostly focus on families of graphs of bounded degree. It turns out that due to the bound on the degree, these collections of natural separations can be partitioned into a bounded number of laminar collections. This in turn allows to us obtain a wide variety of structural and algorithmic results, which we will survey in this talk.

Euiwoong Lee (이의웅), The Karger-Stein algorithm is optimal for k-cut
2021-07-27 15:00 - 16:00 | Room B232, IBS (기초과학연구원)
In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. It is easy to see that the elegant randomized contraction algorithm of Karger and Stein for global mincut (k=2) can be naturally extended for general k with the running time of $O(n^{2k-2})$. Using tree packings, Thorup gave a deterministic algorithm that has the same running time.
In this work, we show that for any fixed $kge 2$, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least $Omega(n^{-k})$. This also gives an extremal bound of $O(n^k)$ on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are essentially tight. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most $(2-delta)OPT/k$, using the Sunflower lemma.
Joint work with Anupam Gupta, David G. Harris, and Jason Li.

Semin Yoo (유세민), Combinatorics of Euclidean spaces over finite fields
2021-07-20 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients, denoted by $binom{n}{k}_{q}$, which give the number of $k$-dimensional subspaces in $mathbb{F}_{q}^{n}$. When $q$ goes to $1$, this reverts to the binomial coefficients which measure the number of $k$-sets in $left [ n right ]$.
In this talk, we add one more structure in $mathbb{F}_{q}^{n}$, which is the Euclidean quadratic form: $text{dot}_{n}:=x_{1}^{2}+x_{2}^{2}+cdots+x_{n}^{2}$. It turns out that the number of quadratic subspaces of Euclidean type in $(mathbb{F}_{q}^{n},text{dot}_{n})$ can be described as the form of the analogue of binomial coefficients. The main goal of this talk is to define the dot-analogues of the binomial coefficients and to study related combinatorics. No prior knowledge about the theory of quadratic form is required.

Stefan Weltge, Integer programs with bounded subdeterminants and two nonzeros per row
2021-07-14 17:00 - 18:00 | Room B232, IBS (기초과학연구원)
We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k=0 (bipartite graphs) and for k=1.
We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to b-matching.
This is joint work with Samuel Fiorini, Gwenaël Joret, and Yelena Yuditsky.

Jaehoon Kim (김재훈), $K_{r+1}$-saturated graphs with small spectral radius
2021-07-13 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $ein E(overline G)$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length 2 in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $nto infty$.

Suil O (오수일), Eigenvalues and [a, b]-factors in regular graphs
2021-07-06 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For positive integers, $r ge 3, h ge 1,$ and $k ge 1$, Bollobás, Saito, and Wormald proved some sufficient conditions for an $h$-edge-connected $r$-regular graph to have a k-factor in 1985. Lu gave an upper bound for the third-largest eigenvalue in a connected $r$-regular graph to have a $k$-factor in 2010. Gu found an upper bound for certain eigenvalues in an $h$-edge-connected $r$-regular graph to have a $k$-factor in 2014. For positive integers $a le b$, an even (or odd) $[a, b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v in V (G)$, $d_H(v)$ is even (or odd) and $a le d_H(v) le b$. In this talk, we provide best upper bounds (in terms of $a, b$, and $r$) for certain eigenvalues (in terms of $a, b, r$, and $h$) in an $h$-edge-connected $r$-regular graph $G$ to guarantee the existence of an even $[a, b]$-factor or an odd $[a, b]$-factor. This result extends the one of Bollobás, Saito, and Wormald, the one of Lu, and the one of Gu.

Florian Gut and Attila Joó, Large vertex-flames in uncountable digraphs
2021-06-30 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
The local connectivity $ kappa_D(r,v) $ from $ r $ to $ v $ is defined to be the maximal number of internally disjoint $rrightarrow v $ paths in $ D $. A spanning subdigraph $ L $ of $ D $ with $ kappa_L(r,v)=kappa_D(r,v) $ for every $ vin V-r $ must have at least $ sum_{vin V-r}kappa_D(r,v) $ edges. It was shown by Lovász that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $ L $ can be captured by the following characterization: For every $ vin V-r $ there is a system $ mathcal{P}_v $ of internally disjoint $ rrightarrow v $ paths in $ L $ covering all the ingoing edges of $ v $ in $ L $ such that one can choose from each $ Pin mathcal{P}_v $ either an edge or an internal vertex in such a way that the resulting set meets every $ rrightarrow v $ path of $ D $. We prove that every digraph of size at most $ aleph_1 $ admits such a spanning subdigraph $ L $. The question if this remains true for larger digraphs remains open.

Jeong Ok Choi (최정옥), Invertibility of circulant matrices of arbitrary size
2021-06-29 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In this talk, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations in terms of the entries in the first row with integer coefficients. Using these conditions we show the invertibility of some family of circulant matrices with particular forms of integers generated by a primitive element in $mathbb{Z}_p$. Also, the invertibility of circulant 0, 1-matrices can be argued combinatorially by applying sufficient conditions. This is joint work with Youngmi Hur.

Hongseok Yang (양홍석), DAG-symmetries and Symmetry-Preserving Neural Networks
2021-06-22 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The preservation of symmetry is one of the key tools for designing data-efficient neural networks. A representative example is convolutional neural networks (CNNs); they preserve translation symmetries, and this symmetry preservation is often attributed to their success in real-world applications. In the machine-learning community, there is a growing body of work that explores a new type of symmetries, both discrete and continuous, and studies neural networks that preserve those symmetries.
In this talk, I will explain what I call DAG-symmetries and our preliminary results on the shape of neural networks that preserve these symmetries. DAG-symmetries are finite variants of DAG-exchangeability developed by Jung, Lee, Staton, and Yang (2020) in the context of probabilistic symmetries. Using these symmetries, we can express that when a neural network works on, for instance, sets of bipartite graphs whose edges are labelled with reals, the network depends on neither the order of elements in the set nor the identities of vertices of the graphs. I will explain how a group of specific DAG-symmetries is constructed by applying a form of wreath product over a given finite DAG. Then, I will explain what linear layers of neural networks preserving these symmetries should look like.
This is joint work with Dongwoo Oh.

Alan Lew, Representability and boxicity of simplicial complexes
2021-06-16 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
An interval graph is the intersection graph of a family of intervals in the real line. Motivated by problems in ecology, Roberts defined the boxicity of a graph G to be the minimal k such that G can be written as the intersection of k interval graphs.
A natural higher-dimensional generalization of interval graphs is the class d-representable complexes. These are simplicial complexes that carry the information on the intersection patterns of a family of convex sets in $mathbb R^d$. We define the d-boxicity of a simplicial complex X to be the minimal k such that X can be written as the intersection of k d-representable complexes.
A classical result of Roberts, later rediscovered by Witsenhausen, asserts that the boxicity of a graph with n vertices is at most n/2. Our main result is the following high dimensional extension of Roberts’ theorem: Let X be a simplicial complex on n vertices with minimal non-faces of dimension at most d. Then, the d-boxicity of X is at most $frac{1}{d+1}binom{n}{d}$.
Examples based on Steiner systems show that our result is sharp. The proofs combine geometric and topological ideas.
