A simple proof of tail--polynomial bounds on the diameter of polyhedra

TL;DR

This paper provides a simplified proof of tail-polynomial bounds on the maximum diameter of polyhedra, improving understanding of polyhedral geometry and potentially impacting optimization algorithms.

Contribution

It offers a unified and simpler analysis of a recursive inequality for polyhedral diameters, leading to new tail-polynomial bounds.

Findings

Simplified proof of tail-polynomial bounds

Unified analysis of recursive inequalities

Improved bounds on polyhedral diameters

Abstract

Let $\Delta(d,n)$ denote the maximum diameter of a $d$-dimensional polyhedron with $n$ facets. In this paper, we propose a unified analysis of a recursive inequality about $\Delta(d,n)$ established by Kalai and Kleitman in 1992. This yields much simpler proofs of a tail--polynomial and tail--almost--linear bounds on $\Delta(d,n)$ which are recently discussed by Gallagher and Kim.