Finding short paths on polytopes by the shadow vertex algorithm

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∈ℝm ×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∈ℤm ×n we show a connection between δ and the largest absolute value Δ of any sub-determinant of A, yielding a bound of O(Δ4mn4) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O(Δ2n4 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(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3 (mn)) by Dyer and Frieze [7].