Approximation algorithm for partial set multicover versus full set multicover

Abstract

In a (full) set multicover (SMC) problem, the goal is to select a subcollection of sets to fully cover all elements, where an element is fully covered if it belongs to at least a required number of sets of the selected subcollection. In a partial set multicover (PSMC) problem, it is sufficient to fully cover a required fraction of elements. In this paper, we present a greedy algorithm for PSMC, and analyze the ratio between the cost of the computed solution and the cost of an optimal solution to the corresponding (full) SMC problem. We shall call this ratio as a partial-versus-full ratio. The motivation for such an analysis is to show that PSMC is fairly economic even when it can only be calculated approximately. It turns out that the partial-versus-full ratio of our algorithm is at most [Formula: see text], where [Formula: see text] is the requirement of an element, and [Formula: see text] is the fraction of elements required to be fully covered. An example is given showing that this ratio is tight up to a constant.