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

DOI: 10.1007/978-3-642-39206-1_24

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)
Cite this paper as:
Brunsch T., Röglin H. (2013) Finding Short Paths on Polytopes by the Shadow Vertex Algorithm. In: Fomin F.V., Freivalds R., Kwiatkowska M., Peleg D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg

## 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 ∈ 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 O4mn4) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O2n4 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].