A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

Abstract

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k u . A solution consists of a function x:V→ℕ0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x u k u . Our objective is to find a cover that minimizes ∑v∈V x v w v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time.