Approximately: Independence Implies Vertex Cover

TL;DR

The paper shows how a near-perfect independent set approximation algorithm can be transformed into a near-perfect vertex cover approximation, leading to new PTAS and QPTAS results for geometric graph problems.

Contribution

It establishes a polynomial-time reduction from approximate Independent Set algorithms to Vertex Cover, enabling new approximation schemes for geometric problems.

Findings

PTAS for Vertex Cover on unweighted pseudo-disks

QQPTAS for Vertex Cover on unweighted axis-aligned rectangles

QPTAS for MWVC on weighted polygons in the plane

Abstract

$\newcommand{\eps}{\varepsilon}$ We observe that a $(1-\eps)$-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a $(1+\eps)$-approximation to Vertex Cover. This basic observation was made before, see [BHR11]. As a consequence, we get a PTAS for VC for unweighted pseudo-disks, QQPTAS for VC for unweighted axis-aligned rectangles in the plane, and QPTAS for MWVC for weighted polygons in the plane. To the best of our knowledge all these results are new.