Complexity of linear relaxations in integer programming

Gennadiy Averkov, Matthias Schymura

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 \((\mathbb {Z}/2\mathbb {Z})^d\), (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.