The role of rationality in integer-programming relaxations

Manuel Aprile, Gennadiy Averkov, Marco Di Summa, Christopher Hojny

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 \({\text {rc}}(X)\) of X as the least number of facets among all possible relaxations Q of X. The rational relaxation complexity \({\text {rc}}_\mathbb {Q}(X)\) restricts the definition of \({\text {rc}}(X)\) 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 \({\text {rc}}(\Delta _d) \le d\) for every \(d \ge 5\). That is, since \({\text {rc}}_\mathbb {Q}(\Delta _d)=d+1\), irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Math Program 154(1):407–425, 2015). Moreover, we prove the asymptotic statement \({\text {rc}}(\Delta _d) \in O(\nicefrac {d}{\sqrt{\log (d)}})\), which shows that the ratio \(\nicefrac {{\text {rc}}(\Delta _d)}{{\text {rc}}_\mathbb {Q}(\Delta _d)}\) goes to 0, as \(d \rightarrow \infty \).