On the approximability of the vertex cover and related problems
Qiaoming Han, Abraham P. Punnen

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 in a graph such that there exists an optimal vertex cover of containing exactly one of the nodes and 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 , where 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 . 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.
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Optimization and Search Problems
