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

We study the partial vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G=(V,E) and an integer s, the 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 vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(V log V + E) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be applied to a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity ku and a weight wu. A solution consists of a function x: V →ℕ0 and an orientation of all but s edges, such that the number edges oriented toward any vertex u is at most xuku. The cost of the cover is given by ∑ v ∈ Vxvwv. Our objective is to find a cover with minimum cost. We provide an algorithm with the same performance guarantee as for regular partial vertex cover. In this case no algorithm for the problem was known.