Parallel approximation for partial set cover

Abstract

In a minimum partial set cover problem (MinPSC), given a ground set E with n elements, a collection S of subsets of E with |S|=m, a cost function c:S→R+, and an integer k≤n, the goal of MinPSC is to find a minimum cost sub-collection of S that covers at least k elements of E. In this paper, we design a parallel algorithm for MinPSC which yields a solution with approximation ratio at most f1−2ε in O(1εlogmnε) rounds, where f is the maximum number of sets containing a common element, and 0<ε<1/2 is a constant. We also design a parallel algorithm for a special MinPSC problem, the minimum power partial cover problem (MinPPC), which achieves approximation ratio at most (3+2ε)α1−2ε in O(1εlogmnεlog2m) rounds, where α≥1 is the attenuation factor of power.