Computational Aspects of Relaxation Complexity: Possibilities and Limitations

TL;DR

This paper explores the computational complexity and bounds of relaxation complexity for integer point sets in polyhedra, providing new algorithms, bounds, and negative results on certifiability.

Contribution

It introduces tight upper bounds for rational relaxation complexity, polynomial-time computation for 2D cases, and investigates the existence of finite certificates for relaxation complexity.

Findings

Polynomial-time computability of rc(X) in 2D.

Existence of sets X with no finite separating set Y.

Explicit formulas for specific classes of X.

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 $\mathrm{rc}(X) \geq k$. 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.