The relaxation complexity of the standard simplex is logarithmic

2026-06-10

AI Summary

This work investigates the relaxation complexity of the discrete standard simplex, defined as the minimum number of facets required by a polyhedron to exactly represent its set of integer points. By employing explicit elementary constructions from combinatorial geometry and polyhedral theory, the authors significantly improve the known upper bound from $O(d / \sqrt{\log d})$ to $O(\log d)$, matching the best-known asymptotic lower bound. This result establishes a tight logarithmic upper bound on the relaxation complexity of the discrete standard simplex, thereby resolving the asymptotic optimality of this fundamental problem in integer programming and polyhedral combinatorics.

Abstract

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $Δ_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(Δ_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(Δ_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

Problem

• relaxation complexity
• standard simplex
• integer points
• polyhedron
• facet complexity

Innovation

• relaxation complexity
• standard simplex
• polyhedral representation
• integer programming
• combinatorial optimization