An update on the Hirsch conjecture

TL;DR

This paper reviews the longstanding Hirsch conjecture in polytope theory, summarizing known results, including proofs and open questions, and discusses the current understanding of polytope diameters relative to the conjecture.

Contribution

It compiles existing results on the Hirsch conjecture, providing accessible proofs and insights into the conjecture's status and related polytope diameter bounds.

Findings

No polynomial upper bound known for polytope diameters

Few polytopes attain the bound n - d

The paper discusses both supporting and opposing evidence

Abstract

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.