On the approximability of the vertex cover and related problems

TL;DR

This paper investigates the complexity of identifying weak edges in graphs related to vertex covers, introduces a new approximation algorithm with performance guarantees, and explores relaxations and LP formulations to improve understanding of vertex cover approximability.

Contribution

It establishes NP-hardness of identifying weak edges, develops a novel approximation algorithm with a specific performance guarantee, and proposes new relaxations and LP models for vertex cover.

Findings

NP-hardness of identifying weak edges in graphs.

A polynomial-time approximation algorithm with performance guarantee 2−1/(1+σ).

New relaxations and LP representations improve understanding of vertex cover approximability.

Abstract

In this paper we show that the problem of identifying an edge $(i,j)$ in a graph $G$ such that there exists an optimal vertex cover $S$ of $G$ containing exactly one of the nodes $i$ and $j$ is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee $2-\frac{1}{1+\sigma}$, where $\sigma$ is an upper bound on a measure related to a weak edge of a graph. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of $\sigma$. We also obtain linear programming representations of the vertex cover problem for special graphs. Our results provide new insights into the approximability of the vertex cover problem - a long standing open problem.