A Modified Greedy Algorithm for the Set Cover Problem with Weights 1 and 2

The set cover problem is that of computing, given a family of weighted subsets of a base set U, a minimum weight subfamily F? such that every element of U is covered by some subset in F?. The k-set cover problem is a variant in which every subset is of size bounded by k. It has been long known that the problem can be approximated within a factor of $$H\left( k \right) = \sum\nolimits_{i = 1}^k {\left( {{1 \mathord{\left/ {\vphantom {1 i}} \right. \kern-\nulldelimiterspace} i}} \right)} $$ 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(k)-1/12 for k-set cover can be attained, when the greedy heuristic is suitably modified for this case.