A modified greedy algorithm for the set cover problem with weights 1 and 2

Abstract

The set cover problem is that of computing, given a family of weighted subsets of a base set U, a minimum weight subfamily F́ such that every element of U is covered by some subset in F́. The κ-set cover problem is a variant in which every subset is of size bounded by κ. It has been long known that the problem can be approximated within a factor of H(κ) = ∑κi=1 (1/i)by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted k-set cover problem, as a first step towards better approximation of general k- set cover problem, where subset costs are limited to either 1 or 2. It will be shown, via LP duality, that improved approximation bounds of H(3)-1/6 for 3-set cover and H(κ)-1/12 for κ-set cover can be attained, when the greedy heuristic is suitably modified for this case. © 2001 Springer Berlin Heidelberg.