Minimum Vertex Cover in Rectangle Graphs

TL;DR

This paper develops advanced algorithms for the Minimum Vertex Cover problem in rectangle intersection graphs, achieving near-optimal solutions for special and general cases by leveraging geometric properties.

Contribution

It introduces an EPTAS for non-crossing rectangles and a 1.5+ε approximation for general rectangles, extending to weighted variants and pseudo-disk intersection graphs.

Findings

EPTAS for non-crossing rectangle families

Approximation factor of (1.5 + ε) for general rectangles

Algorithms exploit geometric properties of rectangles

Abstract

We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families $\calR$ where $R_1 \setminus R_2$ is connected for every pair of rectangles $R_1,R_2 \in \calR$. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of $(1.5 + \varepsilon)$ in general rectangle families, for any fixed $\varepsilon > 0$, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.