On the Shadow Simplex Method for Curved Polyhedra

Dadush, Daniel; Hähnle, Nicolai

doi:10.1007/s00454-016-9793-3

Cited by 27 publications

(39 citation statements)

References 21 publications

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“…In the first submission of our paper, we asked whether an analog of Lemma 2 also holds for linear programs where a local δ-distance property holds. This was answered positively by Dadush and Hähnle [7]. Thus our random walk can be used to solve linear programs whose basis matrices A B , for each feasible basis B, satisfy the δ-distance property in expected polynomial time in n/δ.…”

Section: Remarksmentioning

confidence: 91%

Geometric random edge

Eisenbrand

Vempala

2016

Math. Program.

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We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{c T x : x ∈ R n , Ax b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by n − 1 other rows of A is at least δ.This property is a geometric generalization of A being integral and each sub-determinant of A being bounded by ∆ in absolute value. In this case δ 1/(∆ 2 n). In particular, linear programs defined by totally unimodular matrices are captured in this framework. Here δ 1/n and Dyer and Frieze previously described a strongly polynomial-time randomized simplex algorithm for linear programs with A totally unimodular.The expected number of pivots of the simplex algorithm is polynomial in the dimension and 1/δ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al., using the ideas of Dyer and Frieze. *

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Section: Remarksmentioning

confidence: 91%

Geometric random edge

Eisenbrand

Vempala

2016

Math. Program.

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“…Recently, an improved random walk approach was given by Eisenbrand and Vempala [38], which works in the more general setting where the subdeterminants are bounded in absolute value by \Delta , who gave an O(poly(d, \Delta )) bound on the number of Phase II pivots (note that there is no dependence on n). Furthermore, randomized variants of the shadow vertex algorithm were analyzed in this setting by [24,25], where in particular [25] gave an expected O(d 5 \Delta 2 log(d\Delta )) bound on the number of Phase I and II pivots. Another interesting class of structured polytopes comes from the LPs associated with MDPs, where simplex rules such as Dantzig's most negative reduced cost correspond to variants of policy iteration.…”

Section: Related Workmentioning

confidence: 99%

“…As above, such bounds have been studied for structured classes of polytopes. In particular, the diameter of polytopes with bounded subdeterminants was studied by various authors [37,14,25], where the best known bound of O(d 3 \Delta 2 log(d\Delta )) was given in [25]. The diameters of other classes such as 0/1 polytopes [74], transportation polytopes [8,22,32,21], and flag polytopes [1] have also been studied.…”

Section: Related Workmentioning

confidence: 99%

A Friendly Smoothed Analysis of the Simplex Method

Dadush¹,

Huiberts²

2020

SIAM J. Comput.

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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng [J. ACM, 51 (2004), pp. 385--463] who developed the notion of smoothed analysis. Starting from an arbitrary linear program (LP) with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP, known as the smoothed LP, where variance \sigma 2 Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected \widetil O(d 55 n 86 \sigma - 30 + d 70 n 86 ) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by Deshpande and Spielman [FOCS`05, 2005, pp. 349--356] and later Vershynin [SIAM J. Comput., 39 (2009), pp. 646--678]. The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O \bigl( log 2 n \cdot log log n \cdot (d 3 \sigma - 4 + d 5 log 2 n + d 9 log 4 d)\bigr) number of pivots, improving the dependence on n from polynomial to polylogarithmic. While the original proof of Spielman and Teng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our algorithm requires an expected O(d 2 \surd log n \sigma - 2 + d 3 log 3/2 n) number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with improvements on algorithmic techniques of Vershynin. As an added bonus, our analysis is completely modular and applies to a range of perturbations, which, aside from Gaussians, also includes Laplace perturbations.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . linear programming, shadow vertex simplex method, smoothed analysis \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 52B99, 68Q87, 68W40 \bfD \bfO

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“…Recently, an improved random walk approach was given by Eisenbrand and Vempala [EV17], which works in the more general setting where the subdeterminants are bounded in absolute value by ∆, who gave an O(poly(d, ∆)) bound on the number of Phase II pivots (note that there is no dependence on n). Furthermore, randomized variants of the shadow vertex algorithm were analyzed in this setting by [BGR15,DH16], where in particular [DH16] gave an expected O(d 5 ∆ 2 log(d∆)) bound on the number of Phase I and II pivots. Another interesting class of structured polytopes comes from the LPs associated with Markov Decision Processes (MDP), where simplex rules such as Dantzig's most negative reduced cost correspond to variants of policy iteration.…”

Section: The Cutoff Radiusmentioning

confidence: 99%

A friendly smoothed analysis of the simplex method

Dadush

Huiberts

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM '04), who developed the notion of smoothed analysis. Starting from an arbitrary linear program (LP) with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP, known as the smoothed LP, where variance σ 2 Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected O(d 55 n 86 σ −30 + d 70 n 86 ) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by Deshpande and Spielman (FOCS '05) and later Vershynin (SICOMP '09). The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O log 2 n · log log n · (d 3 σ −4 + d 5 log 2 n + d 9 log 4 d) number of pivots, improving the dependence on n from polynomial to poly-logarithmic.While the original proof of Spielman and Teng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our algorithm requires an expected O(d 2 log n σ −2 + d 3 log 3/2 n) number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with improvements on algorithmic techniques of Vershynin. As an added bonus, our analysis is completely modular and applies to a range of perturbations, which, aside from Gaussians, also includes Laplace perturbations. edges of the feasible polyhedron, where the pivot rule decides which edges to cross, until an optimal vertex or unbounded ray is found. Important examples include Dantzig's most negative reduced cost [Dan51], the Gass and Saaty parametric objective [GS55] and Goldfarb's steepest edge [Gol76] method. We note that for solving LPs in the context of branch & bound and cutting plane methods for integer programming, where the successive LPs are "close together", the dual steepest edge method [FG92] is the dominant algorithm in practice [BFG + 00, Bix12], due its observed ability to quickly re-optimize.The continued success of the simplex method in practice is remarkable for two reasons. Firstly, there is no known polynomial time simplex method for LP. Indeed, there are exponential examples for almost every major pivot rule starting with constructions based on deformed products [KM70, Jer73, AC78, GS79, Mur80, Gol83, AZ98], such as the Klee-Minty cube [KM70], which defeat most classical pivot rules, and more recently based on Markov decision processes (MDP) [FHZ11, Fri11], which notably defeat randomized and history dependent pivot rules. Furthermore, for an LP with d variables and n constraints, the fastest provable (rand...

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On the Shadow Simplex Method for Curved Polyhedra

Abstract: We study the simplex method over polyhedra satisfying certain "discrete curvature" lower bounds, which enforce that the boundary always meets vertices at sharp angles.

Cited by 27 publications

References 21 publications

Geometric random edge

Geometric random edge

A Friendly Smoothed Analysis of the Simplex Method

A friendly smoothed analysis of the simplex method

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