Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

TL;DR

This paper demonstrates that the shadow vertex algorithm can efficiently find short paths on polytopes with bounds related to matrix properties, improving previous bounds especially for totally unimodular matrices.

Contribution

It establishes polynomial bounds on path length and runtime for the shadow vertex algorithm based on polytope flatness and matrix determinants, with improved results for totally unimodular matrices.

Findings

Path length is polynomial in m, n, and 1/delta.

For integer matrices, path length bound depends on the largest sub-determinant Delta.

Special case of totally unimodular matrices yields a significantly improved path length bound.

Abstract

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze.