Minimal Inequalities for an Infinite Relaxation of Integer Programs

We show that maximal S-free convex sets are polyhedra when S is the set of integral points in some rational polyhedron of R. This result extends a theorem of Lovász characterizing maximal lattice-free convex sets. We then consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal S-free convex sets. 1 Maximal S-free convex sets Let S ⊆ Zn be the set of integral points in some rational polyhedron of Rn. We say that B ⊂ Rn is an S-free convex set if B is convex and does not contain any point of S in its interior. We say that B is a maximal S-free convex set if it is convex, and it is not properly contained in any S-free convex set. Our interest in this notion arose from a recent paper of Dey and Wolsey [5] showing the relevance of maximal S-free convex sets in integer programming. When S = Zn, an S-free set is called a lattice-free set. The following theorem of Lovász characterizes maximal lattice-free convex sets. Theorem 1. (Lovász [6]) A set S ⊂ Rn is a maximal lattice-free convex set if and only if one of the following holds: ∗Supported by NSF grant CMMI0653419, ONR grant N00014-03-1-0188 and ANR grant BLAN06-1-138894.