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Subexponential lower bounds for randomized pivoting rules for the simplex algorithm

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

On

doi:10.1145/1993636.1993675
dblp:conf/stoc/FriedmannHZ11
fatcat:rkp3sx46lbc7tpfjz26jkxcagi
*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]

1998
*
Lecture Notes in Computer Science
*

We consider a class A of generalized linear programs on

doi:10.1007/3-540-49543-6_8
fatcat:pmvyoctorze7xir6hz6zblphei
*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

1997
*
Mathematical programming
*

In

doi:10.1007/bf02614318
fatcat:egfjuwbfsvfabomu2fq6ggteua
*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

1995
*
ACM SIGACT News
*

Although many felt t hat t he

doi:10.1145/202840.202847
fatcat:ujieujpfefavxenz5f5qdpppzu
*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]

2011
*
Lecture Notes in Computer Science
*

We provide

doi:10.1007/978-3-642-20807-2_16
fatcat:6umv2rzbxrgcpdgkojtvdc43ki
*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]

2013
*
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Upper

doi:10.1137/1.9781611973402.65
dblp:conf/soda/HansenPZ14
fatcat:vfzwhn62xbgjvgfrtgl3ttr6qy
*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

2013
*
TOP - An Official Journal of the Spanish Society of Statistics and Operations Research
*

I am also grateful to

doi:10.1007/s11750-013-0291-y
fatcat:klst3zgaeva6rmik6ddbdtvf4a
*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

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

doi:10.1145/2746539.2746557
dblp:conf/stoc/HansenZ15
fatcat:hok5knk3dbfmvlckuthnjsip4e
*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

arXiv:1404.3320v1
fatcat:klrycaxktrb5ja4kskkvgdfaia
*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

doi:10.1007/978-3-319-07557-0_2
fatcat:g34vdmujjfesfm7dyolmmk3dhu
*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]

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

2007
*
SIAM Journal on Discrete Mathematics
*

We prove that

doi:10.1137/05062370x
fatcat:376iaqt5mrewpmbvshbsbycebu
*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

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]

2014
*
arXiv
*
pre-print

We then obtained a

arXiv:1410.7871v1
fatcat:dlj3nlyqwjfmlodj7ffpgxnpaq
*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

2006
*
Advances in Mathematics
*

*The*best previous

*lower*

*bound*was quadratic. So in order

*for*

*RANDOM*EDGE to succeed in polynomial time, geometry must help. ... We prove that

*RANDOM*EDGE,

*the*

*simplex*

*algorithm*that always chooses a

*random*improving edge to proceed on, can take a mildly exponential number of steps in

*the*model of abstract objective functions (introduced ... Acknowledgements We would like to thank Ingo Schurr, Uli Wagner and Emo Welzl

*for*stimulating discussions. ...

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