Improved approximation algorithms for the partial vertex cover problem

Abstract

The partial vertex cover problemis a generalization of the vertex cover problem: given an undirected graph G = (V,E) and an integer k, we wish to choose a minimum number of vertices such that at least k edges are covered. Just as for vertex cover, 2-approximation algorithms are known for this problem, and it is of interest to see if we can do better than this. The current-best approximation ratio for partial vertex cover, when parameterized by the maximum degree d of G, is (2−Θ(1/d)). We improve on this by presenting a (formula presented)-approximation algorithm for partial vertex cover using semidefinite programming, matching the current-best bound for vertex cover. Our algorithmuses a new rounding technique, which involves a delicate probabilistic analysis.