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The Complexity of Yamnitsky and Levin’s Simplices Algorithm

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Polytopes — Combinatorics and Computation

Part of the book series: DMV Seminar ((OWS,volume 29))

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Abstract

In 1982 Yamnitsky and Levin gave a variant of the ellipsoid method which uses simplices instead of ellipsoids. Unlike the ellipsoid method this simplices method can be implemented in rational arithmetic. We show, however, that this results in a non-polynomial method since the storage requirement may grow exponentially with the size of the input. Nevertheless, by introducing a rounding procedure we can guarantee polynomiality for both a central-cut and a shallow-cut version. Thus in most applications the simplices method can serve as a substitute for the ellipsoid method. In particular, it performs better than the ellipsoid method if it is used to obtain bounds for the volume of a convex body. Furthermore, it can be used to estimate the optimal function value of total approximation problems.

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Author information

Authors and Affiliations

  1. Emmerich-Josef-Str. 61, Frankfurt/M., D-65929, Germany

    Sven G. Bartels

Authors
  1. Sven G. Bartels
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Editor information

Editors and Affiliations

  1. Institute of Mathematics and Computer Science, The Hebrew University, Givat Ram, 91904, Jerusalem, Israel

    Gil Kalai

  2. Fachbereich Mathematik MA 7-1, Technische Universität Berlin, Strasse des 17. Juni 135, D-10623, Berlin, Germany

    Günter M. Ziegler

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Bartels, S.G. (2000). The Complexity of Yamnitsky and Levin’s Simplices Algorithm. In: Kalai, G., Ziegler, G.M. (eds) Polytopes — Combinatorics and Computation. DMV Seminar, vol 29. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8438-9_10

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  • DOI: https://doi.org/10.1007/978-3-0348-8438-9_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-6351-2

  • Online ISBN: 978-3-0348-8438-9

  • eBook Packages: Springer Book Archive

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