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Discrete & Computational Geometry

, Volume 56, Issue 4, pp 882–909 | Cite as

On the Shadow Simplex Method for Curved Polyhedra

  • Daniel DadushEmail author
  • Nicolai Hähnle
Article

Abstract

We study the simplex method over polyhedra satisfying certain “discrete curvature” lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (Discrete Comput. Geom. 52(1):102–115, 2014), Brunsch and Röglin (Automata, languages, and programming. Part I, pp. 279–290, Springer, Heidelberg, 2013), and Eisenbrand and Vempala (http://arxiv.org/abs/1404.1568, 2014) have improved our understanding of such polyhedra. We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author (in: Indyk P (ed) Proceedings of the Twenty-Sixth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 295–314, SIAM, 2015), who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing. For our first result, we obtain a constructive diameter bound of \(O(\frac{n^2}{\delta } \ln \frac{n}{\delta })\) for n-dimensional polyhedra with curvature parameter \(\delta \in (0,1]\). For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of \(O(n^3 \ln n)\). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected \(O(\frac{n^3}{\delta } \ln \frac{n}{\delta })\) simplex pivots, each requiring O(mn) time to compute, where m is the number of constraints. An initial feasible solution can be found using \(O(\frac{m n^3}{\delta } \ln \frac{n}{\delta })\) pivot steps.

Keywords

Optimization Linear programming Simplex method Diameter of polyhedra 

Notes

Acknowledgments

We would like to thank Friedrich Eisenbrand and Santosh Vempala for useful discussions. We also thank the anonymous reviewers for their useful comments.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Universität BonnBonnGermany

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