Computational aspects of relaxation complexity: possibilities and limitations

Gennadiy Averkov, Christopher Hojny, Matthias Schymura

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 a useful 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. Further, we investigate computable lower bounds on \({{\,\mathrm{rc}\,}}(X)\) with the particular focus on the existence of a finite set \(Y \subseteq \mathbb {Z}^d\) such that separating X and \(Y \setminus X\) allows us to deduce \({{\,\mathrm{rc}\,}}(X) \ge k\). In particular, we show for some choices of X that no such finite set Y exists to certify the value of \({{\,\mathrm{rc}\,}}(X)\), providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for \({{\,\mathrm{rc}\,}}(X)\) for specific classes of sets X and present the first practically applicable approach to compute \({{\,\mathrm{rc}\,}}(X)\) for sets X that admit a finite certificate.