Computer Science > Data Structures and Algorithms
[Submitted on 20 Dec 2007]
Title:On the approximability of the vertex cover and related problems
View PDFAbstract: 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.
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