On sub-determinants and the diameter of polyhedra

TL;DR

This paper establishes new polynomial upper bounds on the diameter of polyhedra based on sub-determinants of the defining matrix, improving previous bounds significantly.

Contribution

It introduces tighter bounds on polyhedron diameters related to sub-determinants, especially for totally unimodular matrices, advancing understanding in polyhedral combinatorics.

Findings

New upper bounds on polyhedron diameter polynomial in n and Δ

Bounds are tighter than previous results by Dyer and Frieze

Special case bounds for totally unimodular matrices

Abstract

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we show that the diameter of P is bounded by O(\Delta^2 n^4 log n\Delta). If P is bounded, then we show that the diameter of P is at most O(\Delta^2 n^3.5 log n\Delta). For the special case in which A is a totally unimodular matrix, the bounds are O(n^4 log n) and O(n^3.5 log n) respectively. This improves over the previous best bound of O(m^16 n^3 (log mn)^3) due to Dyer and Frieze.