Approximation Algorithms for Edge Partitioned Vertex Cover Problems

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We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V,E), a positive cost function c: V-> Z⁺, a partition P₁,..., Pᵣ of the edge set E, and a parameter kᵢ for each partition Pᵢ. The goal is to find a minimum cost set of vertices which cover at least kᵢ edges from the partition Pᵢ. We call this the Partition Vertex Cover problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(log r), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings.