Fractional perfect b-matching polytopes I: General theory

Behrend, Roger E.

doi:10.1016/j.laa.2013.10.001

Cited by 8 publications

(11 citation statements)

References 16 publications

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“…Proof of claim. A proof of the above lemma can be derived from the results given in [4]. We report the details for completeness.…”

Section: Lower Boundmentioning

confidence: 98%

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The Diameter of the Fractional Matching Polytope and Its Hardness Implications

Sanità

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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The (combinatorial) diameter of a polytope P ⊆ R d is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of P , that is the graph where the nodes are given by the 0-dimensional faces of P , and the edges are given the 1-dimensional faces of P . The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard.In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the fractional matching polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an exact characterization of the diameter of the fractional matching polytope, that is of independent interest.

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“…Proof of claim. A proof of the above lemma can be derived from the results given in [4]. We report the details for completeness.…”

Section: Lower Boundmentioning

confidence: 98%

“…Using this result, it is easy to derive adjacency properties for the vertices of P F M . We here explicitly state a lemma that follows from Theorem 25 in [4]. This lemma gives some sufficient conditions for two vertices of P F M to be adjacent.…”

Section: Preliminariesmentioning

confidence: 99%

The Diameter of the Fractional Matching Polytope and Its Hardness Implications

Sanità

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…The adjacency of these fractional polytopes is given in Theorem 25 of [2]. In the following we will only use the fact that the graph of M(G) and PM(G) are, respectively, a subgraph of FM(G) and FPM(G).…”

Section: Monotone and Simplex Paths On Cubes And Zonotopesmentioning

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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“…Let us denote by C(P FMAT (G)) the set of circuits of P FMAT (G) with co-prime integer components. [4,27]), so it remains to be shown that all circuits belong to one of these sets. Let B denote the constraint matrix corresponding to the constraints (2).…”

Section: Key Tool: the Circuits Of The Fractional Matching Polytopementioning

confidence: 99%

“…We first characterize the circuits of the more general fractional matching polytope, i.e., the polytope given by the standard LP-relaxation for the matching problem on general graphs, in Section 3.1. This builds on the known graphical characterization of adjacency given in [27,4]. Then, we construct a reduction from the NP-hard Hamiltonian path problem in Section 3.2 1 .…”

Section: Introductionmentioning

confidence: 99%

Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

Loera¹,

Kafer²,

Sanità³

2019

Preprint

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Circuit-augmentation algorithms are a generalization of the Simplex method, where in each step one is allowed to move along a set of directions, called circuits, that is a superset of the edges of a polytope. We show that in this general context the greatest-improvement and Dantzig pivot rules are NP-hard. Differently, the steepest-descent pivot rule can be computed in polynomial time, and the number of augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1 LPs.Interestingly, we show that this more general framework can be exploited also to make conclusions about the Simplex method itself. In particular, as a byproduct of our results, we prove that (i) computing the shortest monotone path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for 0/1 polytopes, a monotone path of polynomial length can be constructed using steepest improving edges.

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Fractional perfect b-matching polytopes I: General theory

Cited by 8 publications

References 16 publications

The Diameter of the Fractional Matching Polytope and Its Hardness Implications

The Diameter of the Fractional Matching Polytope and Its Hardness Implications

On the Length of Monotone Paths in Polyhedra

Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

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