Computing Kitahara–Mizuno’s bound on the number of basic feasible solutions generated with the simplex algorithm

Kuno, Takahito; Sano, Yoshio; Tsuruda, T.

doi:10.1007/s11590-018-1276-4

Cited by 6 publications

(2 citation statements)

References 13 publications

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“…We remark that the resulting bounds are in general still exponential in the bit-size of the input, and that the constants are complicated to compute. For example, δ is NP-hard to compute in general (see [23]).…”

Section: Together We Havementioning

confidence: 99%

On the Length of Monotone Paths in Polyhedra

Moïse¹,

Loera²,

Louveaux³

2021

SIAM J. Discrete Math.

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Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the TSP, among others.We begin by showing that combinatorial cubes have monotone diameter and Bland simplex height upper bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size monotone diameter. In contrast, we show that for many well-known combinatorial polytopes, the height is at least exponential. Surprisingly, for some famous pivot rules, e.g., greatest improvement and steepest edge, these same polytopes have polynomial-size simplex paths.

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Section: Together We Havementioning

confidence: 99%

On the Length of Monotone Paths in Polyhedra

Moïse¹,

Loera²,

Louveaux³

2021

SIAM J. Discrete Math.

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show abstract

“…We remark that this bound is in general still exponential in the bit-size of the input, and that the constants are complicated to compute. For example, δ is NP-hard to compute in general (see [20]).…”

Section: Monotone and Simplex Paths On Cubes And Zonotopesmentioning

confidence: 99%

On the Length of Monotone Paths in Polyhedra

Moïse¹,

Loera²,

Louveaux³

2020

Preprint

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Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the TSP, among others.We begin by showing that combinatorial cubes have monotone and Bland pivot height bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size pivot height, for all pivot rules. In contrast, we show that many well-known combinatorial polytopes have exponentially-long monotone paths. Surprisingly, for some famous pivot rules, e.g., greatest improvement and steepest edge, these same polytopes have polynomial-size simplex paths.

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On the Number of Degenerate Simplex Pivots

Kukharenko,

Sanità

2024

Lecture Notes in Computer Science

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Computing Kitahara–Mizuno’s bound on the number of basic feasible solutions generated with the simplex algorithm

Cited by 6 publications

References 13 publications

On the Length of Monotone Paths in Polyhedra

On the Length of Monotone Paths in Polyhedra

On the Length of Monotone Paths in Polyhedra

On the Number of Degenerate Simplex Pivots

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