On the Circuit Diameter Conjecture for Counterexamples to the Hirsch Conjecture

TL;DR

This paper investigates whether known counterexamples to the Hirsch conjecture extend to the circuit diameter setting, showing they do not, and introduces structural insights and new orientations related to the conjecture.

Contribution

The study demonstrates that existing Hirsch counterexamples do not transfer to the circuit diameter context and provides new structural observations and orientations that challenge the monotone Hirsch conjecture.

Findings

Counterexamples do not transfer to circuit diameter setting.

Identified four orientations contradicting the monotone Hirsch conjecture.

Analyzed geometric properties of polytopes related to Hirsch counterexamples.

Abstract

Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural question is whether the well-known counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the original counterexamples for the other variants, for bounded polytopes and using monotone walks, also do not transfer. Our results rely on new observations on structural properties of these counterexamples. To resolve the bounded case, we exploit the geometry of certain $2$-faces of the polytopes underlying all known bounded Hirsch counterexamples in Santos' work. For Todd's monotone Hirsch counterexample, we provide two alternative approaches. The first one uses sign-compatible circuit walks, and the second one uses the observation that Todd's polytope is anti-blocking. Along the way, we enumerate all linear programs over the polytope and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining $7107$ satisfy the bound.