TY - GEN

T1 - An improved version of the random-facet pivoting rule for the simplex algorithm

AU - Hansen, Thomas Dueholm

AU - Zwick, Uri

N1 - Publisher Copyright:
© Copyright 2015 ACM.

PY - 2015/6/14

Y1 - 2015/6/14

N2 - 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.

AB - 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.

KW - Linear programming

KW - Randomized pivoting rules

KW - Simplex algorithm

UR - http://www.scopus.com/inward/record.url?scp=84958742708&partnerID=8YFLogxK

U2 - 10.1145/2746539.2746557

DO - 10.1145/2746539.2746557

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AN - SCOPUS:84958742708

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 209

EP - 218

BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

T2 - 47th Annual ACM Symposium on Theory of Computing, STOC 2015

Y2 - 14 June 2015 through 17 June 2015

ER -