Approximation of the Clustered Set Covering Problem

We define a NP-hard clustered variant of the Set Covering Problem where subsets are partitioned into K clusters and a fixed cost is paid for selecting at least one subset in a given cluster. This variant can reformulate as a master problem various multi-commodity flow problems in transportation planning. We show that the problem is approximable within ratio ( 1 + ϵ ) ( e / e − 1 ) H ( q ) , where q is the maximum number of elements covered by a cluster and H ( q ) = ∑ i = 1 q 1 i .