Algorithms column

Abstract

In this column I give a slightly simpler proof of an old result by Nemhauser and Trotter [10]. It would be interesting to see for which other problems such results hold. One of the widely studied problems in Combinatorial Optimization is the Weighted Vertex Cover problem. Given a graph G = (V, E) with a weight function defined on the vertices, a vertex cover is a subset of vertices S such that for each edge e = (u, v) either u E S or v 6 S. The minimum weight vertex cover problem asks for a vertex cover of minimum total weight. This problem has been the subject of many papers, since the problem is NP-hard, but can be solved optimally for bipartite graphs by a reduction to network flows. There are many papers addressing the issue of obtaining polynomial time approximation algorithms [1, 2, 3, 4, 5, 8, 7]. One of the first approaches (in fact this also implies a polynomial time 2 approximation), is by Nemhauser and Trotter [10]. Consider the following simple Integer Program (IP) for the Weighted Vertex Cover Problem. Here Xu refers to a binary indicator variable, and its value is 1 if and only if vertex u is in the cover.