Any Proof of Polynomial Hirsch Must be Completely Incoherent

2026-07-13

AI Summary

This study addresses a coherent-path variant of the polynomial Hirsch conjecture: whether every polytope oriented by a linear objective function admits a monotone path of polynomial length that is consistent with the induced orientation. By integrating fiber polytope theory, directed graph structural analysis, and combinatorial geometric constructions, we explicitly exhibit a family of polytopes together with associated linear functions for which all coherent monotone paths are of exponential length. This result constitutes the first disproof of the conjecture within the framework of coherent monotone paths, demonstrating that approaches relying on coherence cannot resolve the original Hirsch conjecture. Furthermore, it strengthens known lower bounds in the context of the shadow simplex method, geometric transversal problems, and parametric linear optimization.

Abstract

In 1992, Billera and Sturmfels introduced coherent monotone paths on polytopes as part of their description of the fiber polytope construction, and later in 1994 showed with Kapranov that these coherent monotone paths capture the topology of the space of all monotone paths, paths from a minimum to a maximum, in the directed graph of a polytope with orientation induced by a linear function. Those results motivate the following analog of the polynomial Hirsch conjecture: Does there always exist a coherent monotone path of polynomial length on a polytope for any choice of orientation induced by a linear function? We show this is not the case by exhibiting a family of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long. As applications, we strengthen longstanding results pertaining to lower bounds for the shadow simplex method, geometric transversals in discrete geometry, and parametric linear optimization.

Problem

• coherent monotone path
• polynomial Hirsch conjecture
• polytope
• linear function
• exponential length

Innovation

• coherent monotone paths
• polynomial Hirsch conjecture
• fiber polytope
• shadow simplex method
• parametric linear optimization