Complexity of linear relaxations in integer programming

Abstract

AbstractFor a setXof integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with Xis called the relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$rc(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of Xwithout using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding$${{\,\mathrm{rc}\,}}(X)$$rc(X)and its variant$${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rcQ(X), restricting the descriptions of Xto rational polyhedra. As our main results we show that$${{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rc(X)=rcQ(X)when: (a)Xis at most four-dimensional, (b)Xrepresents every residue class in$$(\mathbb {Z}/2\mathbb {Z})^d$$(Z/2Z)d, (c) the convex hull of Xcontains an interior integer point, or (d) the lattice-width of Xis above a certain threshold. Additionally,$${{\,\mathrm{rc}\,}}(X)$$rc(X)can be algorithmically computed when Xis at most three-dimensional, orXsatisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on$${{\,\mathrm{rc}\,}}(X)$$rc(X)in terms of the dimension of X.