KAIST Discrete Math Seminar

All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections. This talk will be non-technical: Nothing will be assumed beyond basic linear algebra.

When network users are satisficing decision-makers, the resulting traffic pattern attains a satisficing user equilibrium, which may deviate from the (perfectly rational) user equilibrium. In a satisficing user equilibrium traffic pattern, the total system travel time can be worse than in the case of the PRUE. We show how bad the worst-case satisficing user equilibrium traffic pattern can be, compared to the perfectly rational user equilibrium. We call the ratio between the total system travel times of the two traffic patterns the price of satisficing, for which we provide an analytical bound. Using the sensitivity analysis for variational inequalities, we propose a numerical method to quantify the price of satisficing for any given network instance.

Let F be a family of convex sets in R^d coloured using d+1 colours. Lovasz’s Colourful Helly Theorem states that if any colourful subfamily of convex sets is intersecting, then one of the monochromatic families is intersecting. We study what happens with the rest of the families.

Approximation algorithms and fixed-parameter tractable (FPT) algorithms have been two major ways to deal with NP-hardness of combinatorial optimization problems. The notion of FPT approximation can be naturally defined, and it is getting significant attention recently. Starting from gentle introductions to approximation algorithms and FPT algorithms, I will talk about my three recent results on FPT approximability.
– Given a graph G = (V, E) and an integer k, we study k-Vertex Separator, where the goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most k vertices. We give an O(log k)-FPT approximation algorithm for k-Vertex Separator. Our result improves the best previous graph partitioning algorithms.
– We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. We present an O(log k)-FPT approximation algorithm for k-Path Transversal. There was no nontrivial approximation algorithm for k > 4 before this work.
– Finally, k-cut is the problem where we want to remove the minimum number of edges such that the graph has at least k connected components. We give a (2 – ε)-FPT approximation algorithm for some epsilon > 0, improving upon a (non-FPT) 2-approximation.
The third result is joint work with Anupam Gupta and Jason Li.

The discrepancy of a sequence f(1), f(2), … of numbers is defined to be the largest value of |f(d) + f(2d) + … + f(nd)| as n,d range over the natural numbers. In the 1930s, Erdős posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwill, as well as a surprising application of the Shannon entropy inequalities, the Erdős discrepancy problem was solved in 2015. In this talk I will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.