Generalized minor inequalities for the set covering polyhedron related to circulant matrices

Abstract

We study the set covering polyhedron related to circulant matrices. In particular, our goal is to characterize the first Chvátal closure of the usual fractional relaxation. We present a family of valid inequalities that generalizes the family of minor inequalities previously reported in the literature and includes new facet-defining inequalities. Furthermore, we propose a polynomial time separation algorithm for a particular subfamily of these inequalities.

On dominating set polyhedra of circular interval graphs

“…In the particular case when A = C k n , facet defining inequalities of Q * (C k n ) related to circulant minors were studied in [2,8,9,29,30].…”

On the Chvátal-rank of facets for the set covering polyhedron of circular matrices

“…Circulant minors of C k n are known to induce valid (and in some cases facetdefining) inequalities for Q * (C k n ). The class of minor inequalities was introduced in [2] and was further studied and generalized in [4,13]. In [5] it was observed that a circulant minor C k n /N ≈ C k n also induces a row family inequality that either is equivalent to or enhances the corresponding minor inequality.…”

“…In [4] it was conjectured that (0, 1)-and (1, 2)-valued minor inequalities suffice to describe Q * (C k n ). This was disproved in [13], where a first example of a facet-defining (2, 3)-valued minor inequality is presented. Using similar ideas as for the case of P * (C k n ) in [8], in this paper we show that there are circulant matrices such that Q * (C k n ) has facet-defining minor related row family inequalities with two consecutive arbitrarily large coefficients.…”