An improved version of the random-facet pivoting rule for the simplex algorithm

Thomas Dueholm Hansen, Uri Zwick

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

Abstract

The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is 2O(√(n-d) log(d/√n-d)), where logn = max{1, logn}. A dual version of the algorithm performs an expected number of at most 2O(√d log ((n-d)/√ d)) dual pivoting steps. This dual version is currently the fastest known combinatorial algorithm for solving general linear programs. Kalai also obtained a primal pivoting rule which performs an expected number of at most 2O(√logn) pivoting steps. We present an improved version of Kalai's pivoting rule for which the expected number of primal pivoting steps is at most min {2O(√(n-d)log (d/(n-d)), 2O(√(d log ((n-d/d))} This seemingly modest improvement is interesting for at least two reasons. First, the improved bound for the number of primal pivoting steps is better than the previous bounds for both the primal and dual pivoting steps. There is no longer any need to consider a dual version of the algorithm. Second, in the important case in which n = O(d), i.e., the number of linear inequalities is linear in the number of variables, the expected running time becomes 2O (√) rather than 2O(√dlogd). Our results, which extend previous results of Gartner, apply not only to LP problems, but also to LP-type problems, supplying in particular slightly improved algorithms for solving 2-player turn-based stochastic games and related problems.

Original languageEnglish
Title of host publicationSTOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages209-218
Number of pages10
ISBN (Electronic)9781450335362
DOIs
StatePublished - 14 Jun 2015
Event47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States
Duration: 14 Jun 201517 Jun 2015

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume14-17-June-2015
ISSN (Print)0737-8017

Conference

Conference47th Annual ACM Symposium on Theory of Computing, STOC 2015
Country/TerritoryUnited States
CityPortland
Period14/06/1517/06/15

Keywords

  • Linear programming
  • Randomized pivoting rules
  • Simplex algorithm

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