Strength of facets for the set covering and set packing polyhedra on circulant matrices

In this paper we prove that the N + -rank coincides with the disjunctive and N-rank for the linear relaxation of the set covering polyhedron of the circulant matrices C s k + 1 k and C s k k if s ⩾ k + 1 . We analyze the behavior of the same operators on the clique relaxation of the stable set polytope of W k 2 with n ⩾ 6 , which has been completely described by Dahl [Dahl, G., Stable Set Polytopes for a Class of Circulant Graphs, SIAM Journal]. We define the strength of the facets with respect to the linear relaxation of the set packing and set covering polyhedra, according to these operators and compare the results with Goemansʹ measure [Argiroffo, G. and S. Bianchi, The nonidealness index of rank-ideal matrices, Preprint (2008)].