# Subexponential lower bounds for randomized pivoting rules for the simplex algorithm

@inproceedings{Friedmann2011SubexponentialLB, title={Subexponential lower bounds for randomized pivoting rules for the simplex algorithm}, author={Oliver Friedmann and Thomas Dueholm Hansen and Uri Zwick}, booktitle={STOC '11}, year={2011} }

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. With essentially all deterministic pivoting rules it is known, however, to require an exponential number of steps to solve some linear programs. No non-polynomial lower bounds were known, prior to this work, for randomized pivoting rules. We provide the first subexponential (i.e., of the form 2Ω(nα), for some α>0) lower bounds for the two most natural, and most studied, randomized pivoting…

## 73 Citations

A Subexponential Lower Bound for Zadeh's Pivoting Rule for Solving Linear Programs and Games

- Computer ScienceIPCO
- 2011

The first subexponential lower bound of the form 2Ω(√n) lower bound is obtained by utilizing connections between pivoting steps performed by simplex-based algorithms and improving switches performed by policy iteration algorithms for 1-player and 2-player games.

DRAFT March 20 , 2012 A subexponential lower bound for the Least Recently Considered rule for solving linear programs and games

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- 2012

The first subexponential lower bound for Cunningham’s Least Recently Considered rule, also known as the ROUND-ROBIN rule, is provided by utilizing connections between pivoting steps performed by simplex-based algorithms and improving switches performed by policy iteration algorithms for 1player and 2-player games.

An Improved Version of the Random-Facet Pivoting Rule for the Simplex Algorithm

- Computer ScienceSTOC
- 2015

An improved version of Kalai's pivoting rule is presented, 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}},)}.

Exponential Lower Bounds for History-Based Simplex Pivot Rules on Abstract Cubes

- Computer Science, MathematicsESA
- 2017

The behavior of the simplex algorithm is a widely studied subject. Specifically, the question of the existence of a polynomial pivot rule for the simplex algorithm is of major importance. Here, we…

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- 2016

It is shown that in the AUSO setting, Random-Edge is slower than Random-Facet, and improves on a 2^{Omega(sqrt^3(n))} lower bound of Matousek and Szabo.

The Polyhedral Geometry of Pivot Rules and Monotone Paths

- Mathematics
- 2022

Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are…

The Complexity of the Simplex Method

- Computer Science, MathematicsSTOC
- 2015

This paper uses the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs to prove that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzes' pivot rule.

A subexponential lower bound for the random facet algorithm for parity games

- Computer ScienceSODA '11
- 2011

This paper focuses in this paper on the algorithm of Matoušek, Sharir and Welzl and refers to it as the Random Facet algorithm, and constructed a family of abstract optimization problems such that the expected running time of the Random facial recognition algorithm, when run on a random instance from this family, is close to the subexponential upper bound.

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- 2021

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the…

The Niceness of Unique Sink Orientations

- Computer Science, MathematicsAPPROX-RANDOM
- 2016

Niceness implies natural upper bounds for Random Edge and it is shown that Random Edge is polynomial on at least $n^{\Omega(2^n)}$ many (possibly cyclic) USO.

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