A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs

Vertex Cover is a fundamental optimization problem, and is among Karp’s 21 NP-complete problems. The problem aims to compute, for a given graph G, a minimum-size set S of vertices of G such that G−S contains no edge. Vertex Cover admits a simple polynomial-time 2-approximation algorithm, which is the best approximation ratio one can achieve in polynomial time, assuming the Unique Game Conjecture. However, on many restrictive graph classes, it is possible to obtain better-than-2 approximation in polynomial time (or even PTASes) for Vertex Cover. In the club of geometric intersection graphs, examples of such graph classes include unit-disk graphs, disk graphs, pseudo-disk graphs, rectangle graphs, etc. In this paper, we study Vertex Cover on the broadest class of geometric intersection graphs in the plane, known as string graphs, which are intersection graphs of any connected geometric objects in the plane. Our main result is a polynomial-time 1.9999-approximation algorithm for Vertex Cover on string graphs, breaking the natural 2 barrier. Prior to this work, no better-than-2 approximation (in polynomial time) was known even for special cases of string graphs, such as intersection graphs of segments. Our algorithm is simple, robust (in the sense that it does not require the geometric realization of the input string graph to be given), and also works for the weighted version of Vertex Cover. Due to a connection between approximation for Independent Set and approximation for Vertex Cover observed by Har-Peled, our result can be viewed as a first step towards obtaining constant-approximation algorithms for Independent Set on string graphs.