Recent progress on the combinatorial diameter of polytopes and simplicial complexes

TL;DR

This paper reviews recent progress on the combinatorial diameter of polytopes and simplicial complexes, including new bounds and collective efforts to understand their maximum diameters, related to the longstanding Hirsch conjecture.

Contribution

It presents a new proof that the maximum diameter of arbitrary simplicial complexes is in $n^{Theta(d)}$, and summarizes collective efforts towards polynomial bounds.

Findings

Maximum diameter of simplicial complexes is in $n^{Theta(d)}$

No polyhedron violating the Hirsch conjecture by more than 25% is known

Collective efforts aim to prove polynomial upper bounds for polytope diameters

Abstract

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in $n^{Theta(d)}$ and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).