The role of rationality in integer-programming relaxations

TL;DR

This paper investigates the relaxation complexity of the vertex set of the standard simplex, showing that irrational relaxations can be significantly smaller than rational ones, and provides asymptotic bounds demonstrating this advantage grows with dimension.

Contribution

It proves that irrational relaxations can reduce the size of relaxations for the simplex vertex set, answering an open question and establishing asymptotic bounds.

Findings

Irrational relaxations can have fewer facets than rational ones for the simplex.

The relaxation complexity of the simplex vertex set is at most linear in the dimension.

The ratio of irrational to rational relaxation complexity tends to zero as dimension increases.

Abstract

For a finite set $X \subset \mathbb{Z}^d$ that can be represented as $X = Q \cap \mathbb{Z}^d$ for some polyhedron $Q$, we call $Q$ a relaxation of $X$ and define the relaxation complexity $rc(X)$ of $X$ as the least number of facets among all possible relaxations $Q$ of $X$. The rational relaxation complexity $rc_\mathbb{Q}(X)$ restricts the definition of $rc(X)$ to rational polyhedra $Q$. In this article, we focus on $X = \Delta_d$, the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in $\mathbb{R}^d$. We show that $rc(\Delta_d) \leq d$ for every $d \geq 5$. That is, since $rc_{\mathbb{Q}}(\Delta_d)=d+1$, irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Lower bounds on the size of integer programs without additional variables, Mathematical Programming, 154(1):407-425, 2015). Moreover, we prove the asymptotic statement $rc(\Delta_d) \in O(\frac{d}{\sqrt{\log(d)}})$, which shows that the ratio $rc(\Delta_d)/rc_{\mathbb{Q}}(\Delta_d)$ goes to $0$, as $d\to \infty$.