Long and winding central paths

TL;DR

This paper constructs linear programs with exponentially curved central paths, disproving a continuous analogue of the Hirsch conjecture by tropicalizing the central path and introducing a tropical angle concept.

Contribution

It introduces a novel tropicalization method for analyzing the central path and provides the first exponential lower bound on its total curvature.

Findings

Central paths can have exponential total curvature.

Tropical geometry offers new tools for analyzing linear programming.

Disproves a continuous analogue of the Hirsch conjecture.

Abstract

We disprove a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with $3r+4$ inequalities in dimension $2r+2$ where the central path has a total curvature in $\Omega(2^r)$. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. The lower bound for the classical curvature is obtained by developing a combinatorial concept of a tropical angle.