Approximation Algorithms for Combinatorial Optimization

Abstract

The partial vertex cover problem is 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 $$\left( {2 - \Theta \left( {\tfrac{{\ln \ln d}}{{\ln d}}} \right)} \right)$$ -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.