Computational Aspects of Relaxation Complexity

Abstract

The relaxation complexity ${{\,\mathrm{rc}\,}}(X)$ of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is an important tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on ${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$, a variant of ${{\,\mathrm{rc}\,}}(X)$ in which the polyhedra P are required to be rational, and we show that ${{\,\mathrm{rc}\,}}(X)$ can be computed in polynomial time if X is 2-dimensional. We also present an explicit formula for ${{\,\mathrm{rc}\,}}(X)$ of a specific class of sets X and present numerical experiments on the distribution of ${{\,\mathrm{rc}\,}}(X)$ in dimension 2.