On the dominating set polytope of web graphs

The dominating set polyhedron of a web graph is the set covering polyhedron of a circulant matrix . Working on the set covering polyhedron of general circulant matrices, Argiroffo and Bianchi found a class of valid inequalities, induced by a particular family of circulant minors. The authors also identified sufficient conditions on the minors in order to induce facet defining inequalities. In this work we generalize these results for new valid inequalities associated with every circulant minor. We conjecture that, for any k, the minor inequalities together with the boolean facets are enough to describe the set covering polyhedron of . This conjecture is true for k=3 as Bouchakour et al. prove working in the context of the Minimum Weight Dominating Set problem (MWDSP) on cycles, i.e, on webs . They also derive the polynomiality of MWDSP on cycles by giving a polynomial separation algorithm for minor inequalities of . Although the computational complexity of the separation problem of every minor inequality is still an open problem, we can state that the set covering problem on matrices is polynomial for a fixed k and obtain polynomial separation algorithms for particular classes of minor inequalities.