The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

Gioan, Emeric; Vergnas, Michel Las

doi:10.1016/j.ejc.2008.12.013

Cited by 12 publications

(90 citation statements)

References 11 publications

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“…If B is the fully optimal basis of H, a direct consequence [4] of the Simplex Criterion is that ∩ b∈B\p b is an optimum of (H; p, f ) for the objective function f being the second element in the minimal basis of H. Hence, computing the fully optimal basis is in fact a more general problem than computing an optimal basis in linear programming.…”

Section: The Fully Optimal Basismentioning

confidence: 97%

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A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

Gioan

Vergnas

2009

Electronic Notes in Discrete Mathematics

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Section: The Fully Optimal Basismentioning

confidence: 97%

“…The main theorem in [4] (see also [1] in the case of graphs) states that if H defines a bounded feasible region, resp. if G is bipolar, w.r.t.…”

Section: The Fully Optimal Basismentioning

confidence: 99%

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A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

Gioan

Vergnas

2009

Electronic Notes in Discrete Mathematics

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“…But the direct single pass algorithm [26] (Theorem 5.8) single pass algorithm [26] (Theorem 5. 8) single pass algorithm [20,24] (Proposition 5.5) deletion/contraction (Theorem 6.13) deletion/contraction (Theorem 6.8) deletion/contraction (Theorem 6.2) full optimality algorithm [29, companion paper] partition of 2 E into activity classes of orientations (Prop. 3.16,Def.…”

Section: Introductionmentioning

confidence: 99%

“…computation is complicated and it had not been addressed in previous papers. When generalized to real hyperplane arrangements, the problem contains and strengthens the real linear programming problem (as shown in [24], hence the name fully optimal). This "one way function" feature is a noteworthy aspect of the active bijection.…”

Section: Introductionmentioning

confidence: 99%

The active bijection for graphs

Gioan

Vergnas

2019

Advances in Applied Mathematics

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The active bijection forms a package of results studied by the authors in a series of papers in oriented matroids. The present paper is intended to state the main results in the particular case, and more widespread language, of graphs. We associate any directed graph, defined on a linearly ordered set of edges, with one particular of its spanning trees, which we call its active spanning tree. For any graph on a linearly ordered set of edges, this yields a surjective mapping from orientations onto spanning trees, which preserves activities (for orientations in the sense of Las Vergnas, for spanning trees in the sense of Tutte), as well as some partitions (or filtrations) of the edge set associated with orientations and spanning trees. It yields a canonical bijection between classes of orientations and spanning trees, as well as a refined bijection between all orientations and edge subsets, containing various noticeable bijections, for instance: between orientations in which smallest edges of cycles and cocycles have a fixed orientation and spanning trees; or between acyclic orientations and no-brokencircuit subsets. Several constructions of independent interest are involved. The basic case concerns bipolar orientations, which are in bijection with their fully optimal spanning trees, as proved in a previous paper, and as computed in a companion paper. We give a canonical decomposition of a directed graph on a linearly ordered set of edges into acyclic/cyclic bipolar directed graphs. Considering all orientations of a graph, we obtain an expression of the Tutte polynomial in terms of products of beta invariants of minors, a remarkable partition of the set of orientations into activity classes, and a simple expression of the Tutte polynomial using four orientation activity parameters. We derive a similar decomposition theorem for spanning trees. We also provide a general deletion/contraction framework for these bijections and relatives. Summary✩ Preprint

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Computing the fully optimal spanning tree of an ordered bipolar directed graph

Gioan,

Las Vergnas

2024

Discrete Mathematics

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The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

Cited by 12 publications

References 11 publications

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The active bijection for graphs

Computing the fully optimal spanning tree of an ordered bipolar directed graph

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