Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

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 $P = \left\{ x \in \mathbb{R}^n \,\colon\, Ax \leq b \right\}$ along the edges of P, where A ∈ ℝ^{m ×n}. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P. For integer matrices A ∈ ℤ^{m ×n} we show a connection between δ and the largest absolute value Δ of any sub-determinant of A, yielding a bound of O(Δ⁴ m n ⁴) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O(Δ² n ⁴ log(n Δ)) by Bonifas et al. [1].

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 log³ (mn)) by Dyer and Frieze [7].

This research was supported by ERC Starting Grant 306465 (BeyondWorstCase).