Archive for the ‘KAIST Discrete Math Seminar’ Category

JinHoo Ahn (안진후), Mekler’s Construction on NTP1 Theory

Tuesday, September 18th, 2018
Mekler’s Construction on NTP1 Theory
JinHoo Ahn (안진후)
Department of Mathematics, Yonsei University, Seoul
2018/10/1 Mon 5PM-6PM (E6-1, Room 1401)
Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler’s construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler’s construction is and briefly show that this preserves NTP1.

Joonkyung Lee (이준경), The extremal number of subdivisions

Thursday, September 13th, 2018
The extremal number of subdivisions
Joonkyung Lee (이준경)
Universität Hamburg, Hamburg, Germany
2018/9/17 Monday 5PM
One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n2 – 1/r edges contains a copy of H. This result is tight up to the constant when H contains a copy of Kr,s with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C4-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and C n3/2 – δ edges contains a copy of H. This answers a question by Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest. This is joint work with David Conlon.

Euiwoong Lee (이의웅), Faster Exact and Approximate Algorithms for k-Cut

Saturday, September 8th, 2018
Faster Exact and Approximate Algorithms for k-Cut
Euiwoong Lee (이의웅)
Courant Institute of Mathematical Sciences, NYU, New York, USA
2018/11/13 Tue 5PM (Bldg. E6-1, Room 1401)
In the k-cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. This problem has been studied various algorithmic perspectives including randomized algorithms, fixed-parameter tractable algorithms, and approximation algorithms. Their proofs of performance guarantees often reveal elegant structures for cuts in graphs.
It has still remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this talk, I will give an overview on recent progresses on both exact and approximation algorithms. Our algorithms are inspired by structural similarities between k-cut and the k-clique problem.

Euiwoong Lee (이의웅), Bridging Continuous and Discrete Optimization through the Lens of Approximation

Saturday, September 8th, 2018

[FYI: Colloquium, School of Computing]

Bridging Continuous and Discrete Optimization through the Lens of Approximation
Euiwoong Lee (이의웅)
Courant Institute of Mathematical Sciences, NYU, New York, USA
2018/11/12 Mon 4PM (Bldg. E3-1, Room 1501)
Mathematical optimization is a field that studies algorithms, given an objective function and a feasible set, to find the element that maximizes or minimizes the objective function from the feasible set. While most interesting optimization problems are NP-hard and unlikely to admit efficient algorithms that find the exact optimal solution, many of them admit efficient “approximation algorithms” that find an approximate optimal solution.
Continuous and discrete optimization are the two main branches of mathematical optimization that have been primarily studied in different contexts. In this talk, I will introduce my recent results that bridge some of important continuous and discrete optimization problems, such as matrix low-rank approximation and Unique Games. At the heart of the connection is the problem of approximately computing the operator norm of a matrix with respect to various norms, which will allow us to borrow mathematical tools from functional analysis.

Jineon Baek, On the off-diagonal Erdős-Szekeres convex polygon problem

Wednesday, August 29th, 2018
On the off-diagonal Erdős-Szekeres convex polygon problem
2018/09/10 Mon 5PM-6PM (Room 1401 of Bldg. E6-1)
The infamous Erdős-Szekeres conjecture, posed in 1935, states that the minimum number ES(n) of points on a plane in general position (that is, no three colinear points) that guarantees a subset of n points in convex position is equal to 2(n-2) + 1. Despite many years of effort, the upper bound of ES(n) had not been better than O(4n – o(n)) until Suk proved the groundbreaking result ES(n)≤2n+o(n) in 2016.
In this talk, we focus on a variant of this problem by Erdős, Tuza and Valtr regarding the number ETV(a, b, n) of points needed to force either an a-cup, b-cap or a convex n-gon for varying a, b and n. They showed ETV(a, b, n) > \sum_{i=n-b}^{a-2} \binom{n}{i-2} by supplying a set of points with no a-cup, b-cap nor a n-gon of that many number, and conjectured that the inequality cannot be improved. Due to their construction, the conjecture is in fact equivalent to the Erdős-Szekeres conjecture. However, even the simplest equality ETV(4, n, n) = \binom{n+1}{2} + 1, which must be true if the Erdős-Szekeres conjecture is, has not been verified yet. To the best of our knowledge, the bound ETV(4, n, n) ≤ \binom{n + 2}{2} – 1, mentioned by Balko and Valtr in 2015, is currently the best bound known in literature.
The talk is divided into two parts. First, we introduce the mentioned works on the Erdős-Szekeres conjecture and observe that the argument of Suk can be directly adapted to prove an improved bound of ETV(a, n, n). Then we show the bound ETV(4, n, n) ≤ \binom{n+2}{2} – C \sqrt{n} for a fixed constant C>0, improving the previously known best bound of Balko and Valtr.