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URN: urn:nbn:de:0030-drops-49125
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### Solving Totally Unimodular LPs with the Shadow Vertex Algorithm

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### Abstract

We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number n of variables, the number m of constraints, and 1/\delta, where \delta is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. n, m, and 1/delta) between two given vertices of a polyhedron [4]. Our result also complements a recent result due to Eisenbrand and Vempala [6] who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables n and 1/\delta, but independent of the number m of constraints. Even though the running time of our algorithm depends on m, it is significantly faster for the important special case of totally unimodular linear programs, for which 1/delta\le n and which have only O(n^2) constraints.

### BibTeX - Entry

@InProceedings{brunsch_et_al:LIPIcs:2015:4912,
author =	{Tobias Brunsch and Anna Gro{\ss}wendt and Heiko R{\"o}glin},
title =	{{Solving Totally Unimodular LPs with the Shadow Vertex Algorithm}},
booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
pages =	{171--183},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-78-1},
ISSN =	{1868-8969},
year =	{2015},
volume =	{30},
editor =	{Ernst W. Mayr and Nicolas Ollinger},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},