The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs

On f-domination: polyhedral and algorithmic results

“…A complete formulation for cycles firstly appears in (Bouchakour et al 2008). This work was extended in (Bianchi et al 2010), leading to complete description for webs of the form W k s(2k+1)+r with s = 2, 3 and 0 ≤ r ≤ s − 1. Also recently, an exact extended formulation for cacti graphs was introduced in (Baïou and Barahona 2014), together with a polynomial-time algorithm to solve [M W 1 ] for cacti.…”

Row family inequalities for the set covering polyhedron

“…For all k, the set covering polyhedron of the circulant matrices C k 2k and C k 3k is described by the inequalities (i) x i ≥ 0 for all i ∈ {1, · · · , n}, (ii) the inequalities from the system C n k x ≥ 1, and (iii) inequalities (6) with n = 1(mod k ).…”

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.…”