Ruth Luo, Induced Turán problems for hypergraphs
2019-11-19 16:30 - 17:30 (Asia/Seoul) | Room B232, IBS (기초과학연구원)
Let $F$ be a graph. We say that a hypergraph $mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $mathcal H$ such that for all $xy in E(F)$, $f(xy) cap V(F) = {x,y}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs. Joint work with Zoltan Füredi.

Frédéric Meunier, Topological bounds for graph representations over any field
2019-11-21 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $mathbb {R}$. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $mathbb {R}$ – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.
This is joint work with Meysam Alishahi.

Jakub Gajarský, First-order interpretations of bounded expansion classes
2019-12-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

Hong Liu, A proof of Mader's conjecture on large clique subdivisions in $C_4$-free graphs
2019-12-12 16:30 - 17:30 | Room 1401, Bldg. E6-1, KAIST
Given any integers $s,tgeq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{frac{1}{2}frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$. This is joint work with Richard Montgomery.

Attila Joó, Base partition for finitary-cofinitary matroid families
2019-12-19 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let ${mathcal{M} = (M_i colon iin K)}$ be a finite or infinite family consisting of finitary and cofinitary matroids on a common ground set $E$.
We prove the following Cantor-Bernstein-type result: if $E$ can be covered by sets ${(B_i colon iin K)}$ which are bases in the corresponding matroids and there are also pairwise disjoint bases of the matroids $M_i$ then $E$ can be partitioned into bases with respect to $mathcal{M}$.

Jaiung Jun (전재웅), The Hall algebra of the category of matroids
2019-12-26 16:30 - 17:30 | Room 1401, Bldg. E6-1, KAIST
To an abelian category A satisfying certain finiteness conditions, one can associate an algebra H_A (the Hall algebra of A) which encodes the structures of the space of extensions between objects in A. For a non-additive setting, Dyckerhoff and Kapranov introduced the notion of proto-exact categories, as a non-additive generalization of an exact category, which is shown to suffice for the construction of an associative Hall algebra. In this talk, I will discuss the category of matroids in this perspective.

Sanjeeb Dash, Boolean decision rules via column generation
2020-01-14 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In many applications of machine learning, interpretable or explainable models for binary classification, such as decision trees or decision lists, are preferred over potentially more accurate but less interpretable models such as random forests or support vector machines. In this talk, we consider boolean decision rule sets (equivalent to boolean functions in disjunctive normal form) as interpretable models for binary classification. We define the complexity of a rule set to be the number of rules (clauses) plus the number of conditions (literals) across all clauses, and assume that simpler or less complex models are more interpretable. We discuss an integer programming formulation for such models that trades off classification accuracy against rule simplicity, and obtain high-quality classifiers of this type using column generation techniques. Compared to some recent alternatives, our algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets, and also produced the winning entry in the 2018 FICO explainable machine learning challenge. When compared to rule learning methods designed for accuracy, our algorithm sometimes finds significantly simpler solutions that are no less accurate.

Ben Lund, Furstenberg sets over finite fields
2020-01-15 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
An important family of incidence problems are discrete analogs of deep questions in geometric measure theory. Perhaps the most famous example of this is the finite field Kakeya conjecture, proved by Dvir in 2008. Dvir’s proof introduced the polynomial method to incidence geometry, which led to the solution to many long-standing problems in the area.
I will talk about a generalization of the Kakeya conjecture posed by Ellenberg, Oberlin, and Tao. A $(k,m)$-Furstenberg set S in $mathbb F_q^n$ has the property that, parallel to every affine $k$-plane V, there is a k-plane W such that $|W cap S| > m$. Using sophisticated ideas from algebraic geometry, Ellenberg and Erman showed that if S is a $(k,m)$-Furstenberg set, then $|S| > c m^{n/k}$, for a constant c depending on n and k. In recent joint work with Manik Dhar and Zeev Dvir, we give simpler proofs of stronger bounds. For example, if $m>2^{n+7}q$, then $|S|=(1-o(1))mq^{n-k}$, which is tight up to the $o(1)$ term.

Adam Zsolt Wagner, The largest projective cube-free subsets of $Z_{2^n}$
2020-01-20 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
What is the largest subset of $Z_{2^n}$ that doesn’t contain a projective d-cube? In the Boolean lattice, Sperner’s, Erdos’s, Kleitman’s and Samotij’s theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $Z_2^n$ we work in $Z_{2^n}$, analogous statements hold if one replaces the word k-chain by projective cube of dimension $2^{k-1}$. The largest d-cube-free subset of $Z_{2^n}$, if d is not a power of two, exhibits a much more interesting behaviour.
This is joint work with Jason Long.

Dillon Mayhew, Courcelle's Theorem for hypergraphs
2020-01-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Courcelle’s Theorem is an influential meta-theorem published in 1990. It tells us that a property of graph can be tested in polynomial time, as long as the property can expressed in the monadic second-order logic of graphs, and as long as the input is restricted to a class of graphs with bounded tree-width. There are several properties that are NP-complete in general, but which can be expressed in monadic logic (3-colourability, Hamiltonicity…), so Courcelle’s Theorem implies that these difficult properties can be tested in polynomial time when the structural complexity of the input is limited.
Matroids can be considered as a special class of hypergraphs. Any finite set of vectors over a field leads to a matroid, and such a matroid is said to be representable over that field. Hlineny produced a matroid analogue of Courcelle’s Theorem for input classes with bounded branch-width that are representable over a finite field.
We have now identified the structural properties of hypergraph classes that allow a proof of Hliněný’s Theorem to go through. This means that we are able to extend his theorem to several other natural classes of matroids.
This talk will contain an introduction to matroids, monadic logic, and tree-automata.
This is joint work with Daryl Funk, Mike Newman, and Geoff Whittle.

Dong Yeap Kang (강동엽), Fragile minor-monotone parameters under random edge perturbation
2020-02-18 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We investigate how minor-monotone graph parameters change if we add a few random edges to a connected graph $H$. Surprisingly, after adding a few random edges, its treewidth, treedepth, genus, and the size of a largest complete minor become very large regardless of the shape of $H$. Our results are close to best possible for various cases. We also discuss analogous results for randomly perturbed bipartite graphs as well as the size of a largest complete odd minor in randomly perturbed graphs.

Xin Zhang (张欣), On the equitable tree-coloring of graphs with low degeneracy
2020-02-25 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A (vertex) $k$-coloring of a graph $G$ is a tree-coloring if each color class induces a forest, and is equitable if the sizes of any two color classes differ by at most 1. The first relative result concerning the equitable tree-coloring of graphs is due to H. Fan, H. A. Kierstead, G. Liu, T. Molla, J.-L. Wu, and X. Zhang (2011), who proved that any graph with maximum degree at most $Delta$ has a $Delta$-coloring so that each color class induces a graph with maximum degree at most 1. After that, many results on this topic were published in the literature. For example, L. Esperet, L. Lemoine, and F. Maffray (2015) showed that any planar graph admits an equitable tree-$k$-coloring for every integer $kge 4$，and G. Chen, Y. Gao, S. Shan, G. Wang, and J.-L. Wu (2017) proved that any 5-degenerate graph with maximum degree at most $Delta$ admits an equitable tree-$k$-coloring for every $kgeq lceilfrac{Delta+1}{2}rceil$.
In this talk, we review part of the known results and the conjectures on the equitable tree-coloring of graphs, and then show the sketch proofs of our three new results as follows:
(a) the vertex set of any graph $G$ can be equitably partitioned into $k$ subsets for any integer $kgeqmax{lceilfrac{Delta(G)+1}{2}rceil,lceilfrac{|G|}{4}rceil}$ so that each of them induces a linear forest;
(b) any plane graph with independent crossings admits an equitable tree-$k$-coloring for every integer $kge 8$;
(c) any $d$-degenerate graph with maximum degree at most $Delta$ admits an equitable tree-$k$-coloring for every integer $kgeq (Delta+1)/2$ provided that $Deltageq 10d$.
This is a joint work with Yuping Gao, Bi Li, Yan Li, and Bei Niu.

Eun-Kyung Cho (조은경), Decomposition of a planar graph into a $d$-degenerate graph and a graph with maximum degree at most $h$
2020-03-03 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Given a graph $G$, a decomposition of $G$ is a collection of spanning subgraphs $H_1, ldots, H_t$ of $G$ such that each edge of $G$ is an edge of $H_i$ for exactly one $i in {1, ldots, t}$. Given a positive integer $d$, a graph is said to be $d$-degenerate if every subgraph of it has a vertex of degree at most $d$. Given a non-negative integer $h$, we say that a graph $G$ is $(d,h)$-decomposable if there is a decomposition of $G$ into two spanning subgraphs, where one is a $d$-degenerate graph, and the other is a graph with maximum degree at most $h$.
It is known that a planar graph is $5$-degenerate, but not always $4$-degenerate. This implies that a planar graph is $(5,0)$-decomposable, but not always $(4,0)$-decomposable. Moreover, by related previous results, it is known that a planar graph is $(3,4)$- and $(2,8)$-decomposable.
In this talk, we improve these results by showing that every planar graph is $(4,1)$-, $(3,2)$-, and $(2,6)$-decomposable. The $(4,1)$- and $(3,2)$-decomposabilities are sharp in the sense that the maximum degree condition cannot be reduced more.
This is joint work with Ilkyoo Choi, Ringi Kim, Boram Park, Tingting Shan, and Xuding Zhu.

Dabeen Lee (이다빈), On a generalization of the Chvátal-Gomory closure
2020-03-17 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Integer programming is the problem of optimizing a linear function over the set of integer solutions satisfying a system of inequalities. The most successful technique in practice is the so-called “cutting-plane” algorithm in combination with branch-and-bound enumeration. Cutting-planes for an integer linear program are linear inequalities that are valid for all integer feasible solutions but cut off intermediate fractional solutions.
The Chvátal-Gomory cuts, introduced by Gomory in 1958 and further studied by Chvátal in 1973 in relation to their applications in combinatorial optimization, are the first class of general-purpose cutting-planes in the literature. The split cuts, whose name was coined by Cook, Kannan, and Schrijver in 1980, are another class of important cutting-planes in modern integer programming. Although there are infinitely many cuts in each class, it is known that only finitely many of them are nonredundant, which is related to designing a finite-convergent cutting-plane algorithm. In this talk, we introduce a new class of cutting-planes that generalizes the Chvátal-Gomory cuts and generalizes a special case of the split cuts. As the two classic classes of cutting-planes, we show that only a finite number of cuts can be redundant.
This talk is based on a joint work with Sanjeeb Dash and Oktay Günlük.

Kevin Hendrey, Covering radius in the Hamming permutation space
2020-03-24 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Our problem can be described in terms of a two player game, played with the set $mathcal{S}_n$ of permutations on ${1,2,dots,n}$. First, Player 1 selects a subset $S$ of $mathcal{S}_n$ and shows it to Player 2. Next, Player 2 selects a permutation $p$ from $mathcal{S}_n$ as different as possible from the permutations in $S$, and shows it to Player 1. Finally, Player 1 selects a permutation $q$ from $S$, and they compare $p$ and $q$. The aim of Player 1 is to ensure that $p$ and $q$ differ in few positions, while keeping the size of $S$ small. The function $f(n,s)$ can be defined as the minimum size of a set $Ssubseteq mathcal{S}_n$ that Player 1 can select in order to gaurantee that $p$ and $q$ will differ in at most $s$ positions.
I will present some recent results on the function $f(n,s)$. We are particularly interested in determining the value $f(n,2)$, which would resolve a conjecture of Kézdy and Snevily that implies several famous conjectures for Latin squares. Here we improve the best known lower bound, showing that $f(n,2)geqslant 3n/4$. This talk is based on joint work with Ian M. Wanless.

Ringi Kim (김린기), The strong clique number of graphs with forbidden cycles
2020-03-31 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The strong clique number of a graph $G$ is the maximum size of a set of edges of which every pair has distance at most two.
In this talk, we prove that every ${C_5,C_{2k}}$-free graph has strong clique number at most $kDelta(G)-(k-1)$, which resolves a conjecture by Cames van Batenburg et al. We also prove that every $C_{2k}$-free graph has strong clique number at most $(2k−1)Delta(G) + (2k−1)^2$, improving the previous known upper bound $10k^2 (Delta(G)-1)$ due to Cames van Batenburg et al. This is joint work with Eun-Kyung Cho, Ilkyoo Choi, and Boram Park.

Pascal Gollin, Disjoint dijoins for classes of dibonds in finite and infinite digraphs
2020-04-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A dibond in a directed graph is a bond (i.e. a minimal non-empty cut) for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of an edge set meeting every dicut equals the maximum number of disjoint dibonds in that digraph. We call such sets dijoins.
Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a digraph equals the minimum size of a dibond.
We study a modification of this question where we restrict our attention to certain classes of dibonds, i.e. whether for a class $mathfrak{B}$ of dibonds of a digraph the maximum number of disjoint edge sets meeting every dibond in $mathfrak{B}$ equal the size a minimum dibond in $mathfrak{B}$.
In particular, we verify this questions for nested classes of dibonds, for the class of dibonds of minimum size, and for classes of infinite dibonds.

Casey Tompkins, Saturation problems in the Ramsey theory of graphs, posets and point sets
2020-04-14 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán’s classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets.
We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdős-Szekeres theorem on convex $k$-gons, as well as multiple semisaturation theorems for sequences and posets.
This project was joint work with Gábor Damásdi, Balázs Keszegh, David Malec, Zhiyu Wang and Oscar Zamora.

Sang-il Oum (엄상일), Survey on vertex-minors
2020-04-21 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For a vertex v of a graph G, the local complementation at v is an operation to obtain a new graph denoted by G*v from G such that two distinct vertices x, y are adjacent in G*v if and only if both x, y are neighbors of v and x, y are non-adjacent, or at least one of x, y is not a neighbor of v and x, y are adjacent. A graph H is a vertex-minor of a graph G if H is obtained from G by a sequence of local complementation and vertex deletions. Interestingly vertex-minors have been used in the study of measurement-based quantum computing on graph states.
Motivated by the big success of the graph minor structure theory developed deeply by Robertson and Seymour since 1980s, we propose a similar theory for vertex-minors. This talk will illustrate similarities between graph minors and graph vertex-minors and give a survey of known theorems and open problems on vertex-minors of graphs.

Seunghun Lee (이승훈), Leray numbers of complexes of graphs with bounded matching number
2020-04-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Given a graph $G$ on the vertex set $V$, the non-matching complex of $G$, $mathsf{NM}_k(G)$, is the family of subgraphs $G’ subset G$ whose matching number $
u(G’)$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $mathsf{NM}_k(K_n)$ and $mathsf{NM}_k(K_{r,s})$ to arbitrary graphs $G$, we show that (i) $mathsf{NM}_k(G)$ is $(3k-3)$-Leray, and (ii) if $G$ is bipartite, then $mathsf{NM}_k(G)$ is $(2k-2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $mathsf{NM}_k(G)$, which vanishes in all dimensions $dgeq 3k-4$, and all dimensions $d geq 2k-3$ when $G$ is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes the result by Aharoni et. al. and Drisko’s theorem: Let $E_1, dots, E_{3k-2}$ be non-empty edge subsets of a graph and suppose that $
u(E_icup E_j)geq k$ for every $i
e j$. Then $E=bigcup E_i$ has a rainbow matching of size $k$. Furthermore, the number of edge sets $E_i$ can be reduced to $2k-1$ when $E$ is the edge set of a bipartite graph.
This is a joint work with Andreas Holmsen.

KSIAM 2020 Spring Conference (cancelled)
2020-05-08 - 2020-05-09 | IBS Science Culture Center
KSIAM 2020 Spring Conference will be held at IBS from May 8, 2020 to May 9, 2020.
Organized by Korean Society for Industrial and Applied Mathematics.
Organizing Committee

Myungjoo Kang (Seoul National University) (chair)
Ahn, Jaemyung (KAIST)
Kwon, Hee-Dae (Inha University)
Lee, Eun Jung (Yonsei University)
Jang, Bongsoo (UNIST)
Jung, Miyoun (Hankuk University of Foreign Studies)
Kim, Junseok (Korea University)
Oum, Sang-il (IBS, KAIST)

Eun Jung Kim (김은정), Twin-width: tractable FO model checking
2020-05-12 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA ’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|phi|) cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS ’15].
In order to explore the limits of twin-width, we generalize to bounded twin-width classes a result by Norine et al. [JCTB ’06] stating that proper minor-free classes are small (i.e., they contain at most $n! c^n$ graphs on $n$ vertices, for some constant $c$). This implies by a counting argument that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width.
Joint work with Stéphan Thomassé, Édouard Bonnet, and Rémi Watrigant.

O-joung Kwon (권오정), Mim-width: a width parameter beyond rank-width
2020-05-19 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Vatshelle (2012) introduced a width parameter called mim-width. It is based on the following cut function : for a vertex partition (A,B) of a graph, the complexity of this partition is computed by the size of a maximum induced matching of the bipartite subgraph induced by edges between A and B. This parameter naturally extends the expressibility power of the graph parameters clique-width and rank-width, which have been well-developed in recent years. In a series of papers, we explored the computational complexity of several problems, parameterized by mim-width. We summarize known structural properties and algorithmic applications of mim-width, and give some open problems at the end. This is joint work with Lars Jaffke, Torstein Strømme, and Jan Arne Telle.

Hong Liu (刘鸿), Asymptotic Structure for the Clique Density Theorem
2020-05-26 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The famous Erdős-Rademacher problem asks for the smallest number of r-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all r was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683-707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for r=3 was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138–160].

Huy-Tung Nguyen, The average cut-rank of graphs
2020-06-02 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The cut-rank of a set X of vertices in a graph G is defined as the rank of the X×(V(G)∖X) matrix over the binary field whose (i,j)-entry is 1 if the vertex i in X is adjacent to the vertex j in V(G)∖X and 0 otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real α, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) α is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) α for each real α≥0. Finally, we describe explicitly all graphs of average cut-rank at most 3/2 and determine up to 3/2 all possible values that can be realized as the average cut-rank of some graph. This is joint work with Sang-il Oum.

Jiseung Kim (김지승), Hardness and concrete security in cryptography
2020-06-09 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Computationally hard problems have been widely used to construct cryptographic primitives such as encryptions, digital signatures. For example, provably secure primitives are based on a reduction from the hardness problems. However, the concrete instantiation of primitives does not follow the results of hardness problems due to its efficiency. In this talk, we introduce cryptographic hardness problems widely used in cryptography and the gap between hardness results and concrete security of cryptographic primitives based on our recent works.

Andreas Holmsen, Fractional Helly and topological complexity
2020-06-16 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The fractional Helly theorem is a simple yet remarkable generalization of Helly’s classical theorem on the intersection of convex sets, and it is of considerable interest to extend the fractional Helly theorem beyond the setting of convexity. In this talk I will discuss a recent result which shows that the fractional Helly theorem holds for families of subsets of $mathbb R^d$ which satisfy only very weak topological assumptions. The proofs combine a number of tools such as homological minors, stair-convexity, supersaturation in hypergraphs, Radon dimension, and Ramsey-type arguments. This is joint work with Xavier Goaoc and Zuzana Patáková.

Jaehoon Kim (김재훈), A resilience version of Pósa's theorem
2020-06-23 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Pósa’s theorem states that any graph G whose degree sequence $d_1leq dots leq d_n$ satisfies $d_i geq i+1$ for all $i< n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs. This is joint work with Padraig Condon, Alberto Espuny Diaz, Daniela Kühn and Deryk Osthus.

Dennis Wong, Generating Gray codes and universal cycles for weak orders
2020-06-30 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A weak order is a way to rank n objects where ties are allowed. Weak orders have applications in diverse areas such as linguistics, designing combination locks, and even in horse racing. In this talk, we present new and simple algorithms to generate Gray codes and universal cycles for weak orders.

Seog-Jin Kim (김석진), Online DP-coloring of graphs
2020-07-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Online list coloring and DP-coloring are generalizations of list coloring that attracted considerable attention recently. Each of the paint number, $chi_P(G)$, (the minimum number of colors needed for an online coloring of $G$) and the DP-chromatic number, $chi_{DP}(G)$, (the minimum number of colors needed for a DP-coloring of $G$) is at least the list chromatic number, $chi_ell(G)$, of $G$ and can be much larger. On the other hand, each of them has a number of useful properties.
We introduce a common generalization, online DP-coloring, of online list coloring and DP-coloring and to study its properties. This is joint work with Alexandr Kostochka, Xuer Li, and Xuding Zhu.
