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**11 - 18**of**18**### Recollections on the discovery of the reverse search technique

"... Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, suppo ..."

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Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, supported by a JSPS/NSERC bilateral exchange. Komei was then working at the University of Tsukuba, Otsuka, which was a couple of subway stations from my office. So we met quite often. One day Komei visited me at my office in Todai and explained to me the criss-cross method for solving linear programs, independently developed by S. Zionts [11], T. Terlaky [8, 9] and Z. Wang [10]. In this method one pivots in the hyperplane arrangement generated by the constraints of the linear program, without regard for feasibility- thus differentiating it from the simplex method. Komei and Tomomi Matsui, a Ph.D. student at Tokyo Institute of Technology, had developed an elegant new proof of the convergence of the criss-cross method [6], which Komei was explaining to me. Komei had drawn a line-arrangement on the blackboard, along with the path the criss-cross method would take from any given vertex to the optimum vertex of the LP. On the board all of these edges were shown in yellow with directions that eventually

### ACImplementation of the Reverse Search Vertex Enumeration Algorithm

, 1993

"... This report documents a C implementation of the reverse search vertex enumeration algorithm for convex polyhedra of Avis and Fukuda. The implementation uses multiple precision rational arithmetic and contains a few improvements over the original algorithm, resulting in a small reduction in the compl ..."

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This report documents a C implementation of the reverse search vertex enumeration algorithm for convex polyhedra of Avis and Fukuda. The implementation uses multiple precision rational arithmetic and contains a few improvements over the original algorithm, resulting in a small reduction in the complexity in most cases. For a polyhedron with n 0 inequalities in d non-negative variables and non-degenerate origin, it finds all bases in time O(n 0d 2)per basis. It is also shown how the implementation can be improved to run in time O(n 0dmin(n 0, d)) time per basis for any polyhedron. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex). Computational experience is included in the report. 1. Background Anew enumeration method called &quot;Reverse Search &quot; was recently introduced by Avis and Fukuda[1] (hereafter referred to as AF) and applied to the problem of finding all vertices of a

### Euclideaness and final polynomials in oriented matroid theory

- COMBINATORICA
, 1993

"... This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda a ..."

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This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids.

### On Circuit Valuation of Matroids

, 2000

"... The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)-valued vectors defined on the circuits of the un ..."

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The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R [ f01g)- valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

### LEAST-INDEX ANTI-CYCLING RULES, LindAcR

, 1998

"... this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel. ..."

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this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel.

### Positroids, non-crossing partitions, and positively oriented matroids

"... Abstract. We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a co ..."

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Abstract. We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals 1/e 2 asymptotically. We also prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. Résumé. Nous étudions le rôle des partitions sans croisements dans l’étude des positroïdes, une classe de matroïdes introduite par Postnikov. On montre que chaque positroïde peut être construit de manière unique par le choix d’une partition sans croisements de l’ensemble [n] ainsi que le choix d’un positroide connexe pour chacun des blocs de la partition. Nous utilisons ce résultat pour énumérer les positroïdes connexes, et nous prouvons que la probabilité qu’un positroïde sur [n] soit connexe est asymptotiquement égale à 1/e 2. Nous prouvons aussi une conjecture de 1987 dûe à da Silva: tout matroïde orienté positivement est un positroïde; autrement dit, il peut être réalisé par un ensemble de vecteurs dans un espace vectoriel réel. Il découle de ce résultat que la Grassmannienne matroïde positive (ou MacPhersonienne positive) est homéomorphe à un boule fermée.