An $11/6$-Approximation Algorithm for Vertex Cover on String Graphs

TL;DR

This paper introduces a simple polynomial-time algorithm achieving an 11/6-approximation for Vertex Cover on string graphs, significantly improving the approximation ratio closer to the trivial bound of 2.

Contribution

It provides the first polynomial-time algorithm with an approximation ratio of 11/6 for Vertex Cover on string graphs, surpassing previous ratios near 2.

Findings

Achieves an 11/6-approximation ratio for Vertex Cover on string graphs.

Establishes that string graphs without small odd cycles are 8-colorable.

Improves the approximation ratio significantly over recent algorithms.

Abstract

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is $11/6$. Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.