Complexity of linear relaxations in integer programming

Abstract

For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity {{,mathrm{rc},}}(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of X without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding {{,mathrm{rc},}}(X) and its variant {{,mathrm{rc},}}_mathbb {Q}(X), restricting the descriptions of X to rational polyhedra. As our main results we show that {{,mathrm{rc},}}(X) = {{,mathrm{rc},}}_mathbb {Q}(X) when: (a) X is at most four-dimensional, (b) X represents every residue class in (mathbb {Z}/2mathbb {Z})^d, (c) the convex hull of X contains an interior integer point, or (d) the lattice-width of X is above a certain threshold. Additionally, {{,mathrm{rc},}}(X) can be algorithmically computed when X is at most three-dimensional, or X satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on {{,mathrm{rc},}}(X) in terms of the dimension of X.