A modified greedy algorithm for dispersively weighted 3-set cover Original Research Article

The set cover problem is that of computing a minimum weight subfamily image, given a family image of weighted subsets of a base set U, such that every element of U is covered by some subset in image. The k-set cover problem is a variant in which every subset is of size at most k. It has been long known that the problem can be approximated within a factor of image 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 3-set cover problem, as a first step towards better approximation of general k-set cover problem, where any two distinct subset costs differ by a multiplicative factor of at least 2. It will be shown, via LP duality, that an improved approximation bound of image can be attained, when the greedy heuristic is suitably modified for this case. A key to our algorithm design and analysis is the Gallai–Edmonds structure theorem for maximum matchings.