Tilted inequalities and facets of the set covering polytope: A theoretical analysis

Abstract

Given a ground-set of elements and a family of subsets, the set covering problem consists in choosing a minimum number of elements such that each subset contains at least one of the chosen elements. This research focuses on the set covering polytope, which is the convex hull of integer solutions to the set covering problem. We investigate the connection between the study of the facets of the set covering polytope and tilting theory. This theory studies how inequalities can be rotated around their contact points with a polyhedron in order to obtain inequalities inducing higher dimensional faces. To study this connection, we introduce the concept of tilting vectors which characterize the degrees of freedom of rotation of an inequality. These vectors characterize facet-defining inequalities and can be used to tilt inequalities with a similar procedure to the one used for arbitrary polyhedra. Additionally, we demonstrate that the computational effort needed to tilt an inequality can be reduced when the inequality has many null coefficients. Finally, we use the tilting vectors to extend several necessary and/or sufficient conditions for facets of the set covering polytope presented by several previous works of the literature.