A Better-Than-Greedy Algorithm for k-Set Multicover

Abstract

The set multicover (MC) problem is a natural extension of the set cover problem s.t. each element requires to be covered a prescribed number of times (instead of just once as in set cover). The k-set multicover (k-MC) problem is a variant in which every subset is of size at most k. Due to the multiple coverage requirement, two versions of MC have been studied; the one in which each subset can be chosen only once (constrained MC) and the other in which each subset can be chosen any number of times (unconstrained MC). For both versions the best approximation algorithm known so far is the classical greedy heuristic, whose performance ratio is H(k), where H(k)= ∑$_{i=1}^{k}$ (1/i). It is no hard, however, to come up with a natural modification of the greedy algorithm such that the resulting performance is never worse, but could also be strictly better. This paper will verify that this is indeed the case by showing that such a modification leads to an improved performance ratio of H(k)–1/6 for both versions of k-MC.