Eun Jung Kim (김은정), New algorithm for multiway cut guided by strong min-max duality

December 17th, 2018

IBS/KAIST Joint Discrete Math Seminar

New algorithm for multiway cut guided by strong min-max duality
Eun Jung Kim (김은정)
CNRS, LAMSADE, Paris, France
2019/01/04 Fri 4PM-5PM (Room: DIMAG, IBS)

Problems such as Vertex Cover and Multiway Cut have been well-studied in parameterized complexity. Cygan et al. 2011 drastically improved the running time of several problems including Multiway Cut and Almost 2SAT by employing LP-guided branching and aiming for FPT algorithms parameterized above LP lower bounds. Since then, LP-guided branching has been studied in depth and established as a powerful technique for parameterized algorithms design.

In this talk, we make a brief overview of LP-guided branching technique and introduce the latest results whose parameterization is above even stronger lower bounds, namely μ(I)=2LP(I)-IP(dual-I). Here, LP(I) is the value of an optimal fractional solution and IP(dual-I) is the value of an optimal integral dual solution. Tutte-Berge formula for Maximum Matching (or equivalently Edmonds-Gallai decomposition) and its generalization Mader’s min-max formula are exploited to this end. As a result, we obtain an algorithm running in time 4k-μ(I)for multiway cut and its generalizations, where k is the budget for a solution.

This talk is based on a joint work with Yoichi Iwata and Yuichi Yoshida from NII.

Hong Liu, Polynomial Schur’s Theorem

December 4th, 2018

IBS/KAIST Joint Discrete Math Seminar

Polynomial Schur’s Theorem
Hong Liu
University of Warwick, UK
2018/12/13 Thu 5PM-6PM (Room B109, IBS)
I will discuss the Ramsey problem for {x,y,z:x+y=p(z)} for polynomials p over ℤ. This is joint work with Peter Pach and Csaba Sandor.

Tony Huynh, A tight Erdős-Pósa function for planar minors

December 4th, 2018

IBS/KAIST Joint Discrete Math Seminar

A tight Erdős-Pósa function for planar minors
Tony Huynh
Université libre de Bruxelles
2018/12/10 5PM-6PM (Room B109, IBS)
Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f(k) such that for all k and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G−X has no H-minor. We prove that this remains true with f(k)=ck log k for some constant c depending on H. This bound is best possible, up to the value of c, and improves upon a recent bound of Chekuri and Chuzhoy. The proof is constructive and yields the first polynomial-time O(log ???)-approximation algorithm for packing subgraphs containing an H-minor.

This is joint work with Wouter Cames van Batenburg, Gwenaël Joret, and Jean-Florent Raymond.

Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs

October 31st, 2018
Largest 2-regular subgraphs in 3-regular graphs
Ilkyoo Choi (최일규)
Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si
2018/11/26 Mon 5PM-6PM (Bldg. E6-1, Room 1401)
For a graph G, let f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f2(G) over 3-regular n-vertex simple graphs G.
To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.
More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.
These bounds are sharp; we describe the extremal multigraphs.
This is joint work with Ringi Kim, Alexandr V. Kostochka, Boram Park, and Douglas B. West.

Jaehoon Kim, Introduction to Graph Decomposition

October 2nd, 2018
Introduction to Graph Decomposition
Jaehoon Kim (김재훈)
Mathematics Institute, University of Warwick, UK
2018/10/15 5PM
Graphs are mathematical structures used to model pairwise relations between objects.
Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.
In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.