A Proof of the Strict Monotone 5-Step Conjecture

Gallagher, J. Mackenzie; Morris, Walter D.

doi:10.1007/s00454-019-00161-3

Cited by 3 publications

(2 citation statements)

References 12 publications

(244 reference statements)

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“…Our results imply that this bound also holds for oriented matroid programs. See [12] for a recent application of oriented matroid programming to determination of polytope diameters. In dimension 3, the Holt-Klee condition is also known to be sufficient for an acyclic orientation of the graph of a polytope with a unique source and sink on every face to be d-polytopal.…”

Section: Introductionmentioning

confidence: 99%

On the Holt-Klee Property for Oriented Matroid Programming

Morris¹

2021

Preprint

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The Holt-Klee theorem says that the graph of a d-polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d + 1 on n + 2 elements, which include those from d-polytopes with n facets, admit d independent monotone paths from source to sink if d ≤ 4. This was previously only known to hold for d ≤ 3 and n ≤ 6.

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Section: Introductionmentioning

confidence: 99%

On the Holt-Klee Property for Oriented Matroid Programming

Morris¹

2021

Preprint

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“…the TSP [30,31,32]. Today, several papers continue the study of shortest monotone paths (see [8,13,29] and references therein).…”

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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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On the Holt–Klee property for oriented matroid programming

Morris

2022

European Journal of Combinatorics

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A Proof of the Strict Monotone 5-Step Conjecture

Cited by 3 publications

References 12 publications

On the Holt-Klee Property for Oriented Matroid Programming

On the Holt-Klee Property for Oriented Matroid Programming

On the Length of Monotone Paths in Polyhedra

On the Holt–Klee property for oriented matroid programming

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