ICALP 2013: Automata, Languages, and Programming pp 279-290

# Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

• Tobias Brunsch
• Heiko Röglin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

## 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 O4 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 O2 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].

## Keywords

Simplex Algorithm Compute Path Integer Matrice Pivot Rule Unimodular Matrice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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