Random-edge is slower than random-facet on abstract cubes

Thomas Dueholm Hansen, Uri Zwick

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2Ω(√n log n) steps. This improves on a 2√(3√ n) lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2O(√n) steps on any n-dimensional AUSO, this established our result. Improving our 2Ω(√n log n) lower bound seems to require radically new techniques.

Original languageEnglish
Title of host publication43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
EditorsYuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770132
DOIs
StatePublished - 1 Aug 2016
Event43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy
Duration: 12 Jul 201615 Jul 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume55
ISSN (Print)1868-8969

Conference

Conference43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
Country/TerritoryItaly
CityRome
Period12/07/1615/07/16

Keywords

  • Acyclic unique sink orientations
  • Linear programming
  • Pivoting rules
  • The simplex algorithm

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