### G-2003-44

# Solution Techniques for the Large Set Covering Problem

## Philippe Galinier and Alain Hertz

Given a finite set *E* and a family *F*={*E*_{1},...,*E _{m}*} of subsets
of

*E*such that

*F*covers

*E*, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of

*F*that also covers

*E*. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that

*E*and the subsets

*E*are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of

_{i}*F*that does not cover

*E*. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the

*k*-colouring problem which is formulated as a constraint satisfaction problem.

Published **July 2003**
,
22 pages