Lift-and-project ranks of the set covering polytope of circulant matrices

Abstract

In this paper, we analyze the behavior of the N and N+ operators (defined by Lovász and Schrijver) and the disjunctive operator due to Balas, Ceria and Cornuéjols, on the linear relaxation of the set covering polytope associated with circulant matrices Cnk. We found that for the family of circulant matrices Csk+1k the disjunctive rank coincides with the N- and N+-rank at the value k−1. This result provides bounds for lift-and-project ranks of most circulant matrices since Csk+1k appears as a minor of almost all circulant matrices. According to these operators, we define the strength of facets with respect to the linear relaxation of the set covering polytope and compare the results with a similar measure previously defined by Goemans. We identify facets of maximum strength although the complete description of the set covering polytope of circulant matrices is still unknown. Moreover, considering the matrices Cskk with s≥k+1, we found a family of facets of the corresponding set covering polyhedron, having maximum strength according to the disjunctive and Goemans’ measures.