Diameters of Cocircuit Graphs of Oriented Matroids: An Update

TL;DR

This paper updates the bounds on the diameter of cocircuit graphs in oriented matroids, with new exact bounds for low-rank cases and computational results for small matroids, relevant to optimization algorithms.

Contribution

It provides the latest bounds on cocircuit graph diameters, reduces the problem to uniform cases, and improves existing bounds, advancing understanding in combinatorics and optimization.

Findings

Diameter bounds for low-rank oriented matroids

Exact bounds for all oriented matroids with up to nine elements

Improved quadratic bound for arbitrary oriented matroids

Abstract

Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). The motivation for our investigations is the complexity of the simplex method and the criss-cross method. For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights two very important conjectures related to the polynomial Hirsch conjecture for polytope diameters.