Solving Totally Unimodular LPs with the Shadow Vertex Algorithm

TL;DR

This paper demonstrates that the shadow vertex simplex algorithm can solve totally unimodular linear programs in strongly polynomial time, improving efficiency for this important class of problems by leveraging geometric properties.

Contribution

It extends the shadow vertex algorithm's applicability to strongly polynomial time solutions for totally unimodular LPs, combining geometric insights with algorithmic analysis.

Findings

Strongly polynomial time solution for totally unimodular LPs

Shadow vertex algorithm's efficiency depends on geometric flatness parameter

Significant speed-up for LPs with O(n^2) constraints

Abstract

We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number $n$ of variables, the number $m$ of constraints, and $1/\delta$, where $\delta$ is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. $n$, $m$, and $1/\delta$) between two given vertices of a polyhedron. Our result also complements a recent result due to Eisenbrand and Vempala who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables $n$ and $1/\delta$, but independent of the number $m$ of constraints. Even though the running time of our algorithm depends on $m$, it is significantly faster for the important special case of totally unimodular linear programs, for which $1/\delta\le n$ and which have only $O(n^2)$ constraints.