A note on the Clustered Set Covering Problem

We define an 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. We show that the problem is approximable within ratio (1+ ǫ)(e/e− 1)H(q), where q is the maximum number of elements covered by a cluster and H(q) = ∑q i=1 1 i . Key-words: Integer Programming, Set Covering, Maximal Coverage, Approximation. 1 Problem statement In the classical Set Covering Problem (SCP), we are given a set of elements C = {1, . . . , n}, a collection S = {S1, . . . , Sm} ⊆ 2 C of subsets of C covering C and a non-negative weight c(Sj) ≥ 0 for each set Sj ∈ S. The goal is to find a set cover S ′ = {Sj1, . . . , Sjt} ⊆ S, verifying ∪l=1Sjl = C, and minimizing c(S ′) = ∑t l=1 c(Sjl). This problem has been widely studied by the computer science community and the main results given on it are the following: SCP is NP-hard, even in the unweighted case, i.e., c(Sj) = 1 ∀j = 1, . . . ,m [8]. SCP is H(∆)-approximable where H(∆) = ∑∆ i=1 1 i and ∆ is the maximum size of a set of S, i.e., ∆ = maxj≤m |Sj| [4]; this gives a (1 + lnn)-approximation for SCP since ∆ ≤ n and H(n) ≤ 1 + lnn. On the other hand, SCP is not (1 − ε) ln n-approximable for every ε > 0 [7] closing the gap between positive and negative results on this problem. Finally, the restriction of SCP where ∆ and δ are upper bounded by some constants is APX-complete [12]; here δ is the maximum number of sets of S containing a given element of C, i.e., δ = max{p : ∃Sj1, . . . , Sjp such that ∩ p l=1Sjl 6= ∅}. We define the following variant of SCP, called Clustered Set Covering Problem (ClusteredSCP). Let C = {1, . . . , n} be a set of elements and S = {S1, . . . , Sm} be a collection of subsets of C. A positive cost cj = c(Sj) is associated with every subset Sj ∈ S. Moreover, we assume that the index set J = {1, . . . ,m} is partitioned into K disjoint subsets J = {Jk : k = 1, . . . ,K}, i.e., ∪ K k=1Jk = J and Jk ∩ Jk′ = ∅ for k 6= k ′. For k = 1, . . . ,K, cluster Fk ⊂ S is defined by Fk = {Sj ∈ S : j ∈ Jk}, and a fixed-cost fk ≥ 0 is paid as soon as at least one subset is selected within cluster Fk for k = 1, . . . ,K. The Clustered Set Covering Problem is to cover all elements of C by a collection of subsets S ′ ⊂ S minimizing the sum of the costs of the selected subsets and the fixed costs. In other words, we want to find a set cover S ′ minimizing c(S ′) plus the cost of the clusters used in S ′, i.e., Research supported by the French Agency for Research under the DEFIS program TODO, ANR-09EMER-010. ESSEC Business School, Av. B. Hirsch, 95021 Cergy Pontoise, France, [email protected] CNRS, UMR 7243, F-75775 Paris, France, and Université Paris-Dauphine, LAMSADE, F-75775 Paris, France, [email protected]