Approximately: Independence Implies Vertex Cover

TL;DR

The paper links the quality of approximating Vertex Cover to that of approximating Independent Set by a dense-subgraph reduction, showing that a $(1-\varepsilon)$-approximation to IS on all induced subgraphs yields a $(1+\varepsilon)$-approximation to VC in polynomial time. The core method hinges on a half-integral LP, crown decompositions, and a reduction to a dense subgraph $K=G[V_{1/2}]$, where the LP assigns value $\tfrac{1}{2}$ to all vertices. The main theorem then leverages an induced-subgraph IS approximation algorithm to produce a near-optimal VC, with concrete PTAS/QQPTAS/QPTAS consequences for several geometric intersection graphs. This framework yields practical, near-optimal VC algorithms for unweighted pseudo-disks, axis-aligned rectangles, and weighted polygons in the plane, highlighting a broad bridge between IS and VC in geometric settings.

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.

Theorems & Definitions (7)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: Nemhauser-Trotter
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Theorem 3.2