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

Oliver Friedmann, Thomas Dueholm Hansen, Uri Zwick
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
On lower bounds for the RandomFacet pivoting rule and the Randomized Bland's rule.  ...  Lower bounds for the simplex algorithm utilizing Markov decision processes. Example: A lower bound for Bland's rule. Gadgets and general ideas for the lower bound for RandomEdge.  ...  bounds for the RandomFacet pivoting rule and the Randomized Bland's rule.  ...

### Combinatorial Linear Programming: Geometry Can Help [chapter]

Bernd Gärtner
1998 Lecture Notes in Computer Science
We consider a class A of generalized linear programs on the d-cube (due to Matou sek) and prove that Kalai's subexponential simplex algorithm Random-Facet is polynomial on all actual linear programs in  ...  In contrast, the subexponential analysis is known to be best possible for general instances in A.  ...  Acknowledgment I would like to thank Jirka Matou sek, Falk Tschirschnitz, Emo Welzl and G unter Ziegler for many remarks that helped to improve the presentation.  ...

### Linear programming, the simplex algorithm and simple polytopes

Gil Kalai
1997 Mathematical programming
In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes.  ...  In the rst part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes.  ...  And we cannot nd a deterministic pivot rule (without randomization) which is not exponential. We leave these tasks for you the reader.  ...

### A survey of linear programming in randomized subexponential time

Michael Goldwasser
1995 ACM SIGACT News
Although many felt t hat t he simplex method was polynomial in the w orst case, Klee and Minty provided counterexamples in 1972, requiring exponential time for a common pivot rule 13 .  ...  By using a s t andard simplex algorithm as a subroutine for solving t hese small programs, Clarkson's algorithm is still exponential, however the best current b o u nds are realized by u s i n g o n e  ...  Acknowledgments: The a uthor wishes to t hank Rajeev Motwani, Leo Guibas and Bernd G artner for their helpful discussions.  ...

### A Subexponential Lower Bound for Zadeh's Pivoting Rule for Solving Linear Programs and Games [chapter]

Oliver Friedmann
2011 Lecture Notes in Computer Science
We provide the first subexponential (i.e., of the form 2 Ω( √ n ) lower bound for this rule.  ...  The simplex algorithm is among the most widely used algorithms for solving linear programs in practice.  ...  I would like to thank Uri Zwick and Thomas Dueholm Hansen for pointing me to this challenging pivoting rule and for numerous inspiring discussions on the subject.  ...

### Improved upper bounds for Random-Edge and Random-Jump on abstract cubes [chapter]

Thomas Dueholm Hansen, Mike Paterson, Uri Zwick
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
Upper bounds are given for the complexity of two very natural randomized algorithms for finding the sink of an Acyclic Unique Sink Orientation (AUSO) of the ncube.  ...  For Random-Edge, we obtain an upper bound of about 1.80 n , improving upon the the previous upper bound of about 2 n /n log n obtained by Gärtner and Kaibel.  ...  Can the exponential upper bound for Random-Edge be improved to a subexponential upper bound, or can the subexponential lower bound of Matoušek and Szabó [17] be improved to a genuinely exponential lower  ...

### Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Jesús A. De Loera
2013 TOP - An Official Journal of the Spanish Society of Statistics and Operations Research
I am also grateful to the Technische Universität München for the hospitality received during the time of writing this article.  ...  I also want to thank the editors of this volume for the invitation to contribute a commentary to this special issue.  ...  The team of Friedmann et al. (2011) provided the first lower bound of the form 2 Ω(n α ) , for some α > 0, for both the Random-Edge and the Random-Facet pivot rule in the one-pass variant.  ...

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

Thomas Dueholm Hansen, Uri Zwick
2015 Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
Kalai [33] and Matoušek, Sharir and Welzl [46] devised a randomized pivoting rule, Random-Facet, for the simplex algorithm and obtained a subexponential 2 O( (n−d) log (d/ √ n−d) ) upper bound on the expected  ...  Random-Facet is currently the fastest known pivoting rule for the simplex algorithm.  ...  We believe that the improved pivoting rule is essentially the best pivoting rule that can be obtained using the (simple and ingenious) idea of choosing a random facet.  ...

### On Simplex Pivoting Rules and Complexity Theory [article]

2014 arXiv   pre-print
We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path.  ...  Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We conjecture that the same can be shown for most known variants of the simplex method.  ...  The simplest one [9] is to pick a random index in J(B). Another important class of randomized rules are the random facet rules used in the proofs of subexponential diameter bounds [16, 17, 24] .  ...

### On Simplex Pivoting Rules and Complexity Theory [chapter]

2014 Lecture Notes in Computer Science
We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path.  ...  Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We conjecture that the same can be shown for most known variants of the simplex method.  ...  The simplest one [9] is to pick a random index in J(B). Another important class of randomized rules are the random facet rules used in the proofs of subexponential diameter bounds [16, 17, 24] .  ...

### Linear programming — Randomization and abstract frameworks [chapter]

Bernd Gärtner, Emo Welzl
1996 Lecture Notes in Computer Science
The bound relies on two algorithms by Clarkson, and the subexponential algorithms due to Kalai, and to Matou sek, Sharir & Welzl. We review some of the recent algorithms with their analyses.  ...  Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized'combinatorial' algorithm  ...  For many pivot rules, the simplex method was shown to require exponential time on certain inputs (the rst such input has been constructed by Klee and Minty 19] for the pivot rule originally proposed  ...

### Two New Bounds for the Random‐Edge Simplex‐Algorithm

Bernd Gärtner, Volker Kaibel
2007 SIAM Journal on Discrete Mathematics
We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices.  ...  As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2^d on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.  ...  randomized pivot rules.  ...

### Randomized Simplex Algorithms on Klee-Minty Cubes

Bernd Gärtner, Martin Henk, Günter M. Ziegler
1998 Combinatorica
The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots.  ...  At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation.  ...  We are indebted to Jiří Matoušek for a substantial simplification of the lower bound proof in Section 3. Finally, Noga Alon contributed to the improved upper bound of Section 3.  ...

### Errata for: A subexponential lower bound for the Random Facet algorithm for Parity Games [article]

Oliver Friedmann and Thomas Dueholm Hansen and Uri Zwick
2014 arXiv   pre-print
We then obtained a lower bound on the expected number of pivoting steps performed by Random-Facet^* and claimed that the same lower bound holds also for Random-Facet.  ...  In Friedmann, Hansen, and Zwick (2011) we claimed that the expected number of pivoting steps performed by the Random-Facet algorithm of Kalai and of Matousek, Sharir, and Welzl is equal to the expected  ...  of pivoting steps performed by Random-Facet and Random-Facet * are not the same.  ...

### RANDOM EDGE can be exponential on abstract cubes

Jiří Matoušek, Tibor Szabó