Book•
Graphes et algorithmes
01 Jan 1995-
TL;DR: Les algorithmes approches / Annexes :Programmation lineaire - Programmation LINEAIR en nombres entiers - Relaxation lagrangienne et resolution du probleme dual - Programme dynamique - Les problemes de ratio m minimumm as discussed by the authors.
Abstract: 1.Generalites sur les graphes / 2.Le probleme du plus court chemin / 3. Les algebres de chemins / 4.Arbres et arborescences / 5. Flots et reseaux de transport / 6.Flots avec multiplicateurs.multiflots / 7.Couplages et b-couplages / 8.Parcours euleriens et hamiltoniens / 9.Matroides / 10.Les problemes"non polynomiaux"/ 11.Les algorithmes d'enumeration par separation et evaluations / 12.Les algorithmes approches / Annexes :Programmation lineaire - Programmation lineaire en nombres entiers - Relaxation lagrangienne et resolution du probleme dual - Programmation dynamique - Les problemes de ratio m minimumm
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TL;DR: The classical shortest paths algorithms of the graph theory are revisited in order to derive new watershed algorithms, which are either new or more easy to implement in hardware.
1,355 citations
TL;DR: In this paper, a solution procedure for the Elementary Shortest Path Problem with Resource Constraints (ESPPRC) is proposed, which extends the classical label correcting algorithm originally developed for the relaxed (nonelementary) path version of this problem.
Abstract: In this article, we propose a solution procedure for the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). A relaxed version of this problem in which the path does not have to be elementary has been the backbone of a number of solution procedures based on column generation for several important problems, such as vehicle routing and crew pairing. In many cases relaxing the restriction of an elementary path resulted in optimal solutions in a reasonable computation time. However, for a number of other problems, the elementary path restriction has too much impact on the solution to be relaxed or might even be necessary. We propose an exact solution procedure for the ESPPRC, which extends the classical label correcting algorithm originally developed for the relaxed (nonelementary) path version of this problem. We present computational experiments of this algorithm for our specific problem and embedded in a column generation scheme for the classical Vehicle Routing Problem with Time Windows. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 216–229 2004
606 citations
19 Feb 2003
TL;DR: This paper addresses the forbidden state problem of Petri nets with liveness requirement and uncontrollable transitions with the proposed approach computes a maximally permissive PN controller, whenever such a controller exists.
Abstract: This paper addresses the forbidden state problem of Petri nets (PN) with liveness requirement and uncontrollable transitions. The proposed approach computes a maximally permissive PN controller, whenever such a controller exists. The first step, based on a Ramadge-Wonham-like reasoning (1989), determines the legal and live maximal behavior the controlled PN should have. In the second step, the theory of regions is used to design control places to add to the original model to realize the desired behavior. Furthermore, necessary and sufficient conditions for the existence of control places realizing the maximum permissive control are given. A parameterized manufacturing application of significant state space is used to show the efficiency of the proposed approach.
333 citations
TL;DR: The notion of geodesic neighborhood is introduced in order to define some topological numbers associated with a point in a three-dimensional cubic grid which lead to a characterization of simple points which consists in only two local conditions.
280 citations
27 Feb 1997
TL;DR: Exotic semirings have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis.
Abstract: Exotic semirings such as the “(max, +) semiring” (ℝ ∪ {−∞},max,+), or the “tropical semiring” (ℕ ∪ {+∞}, min, +), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities).
243 citations
Cites background from "Graphes et algorithmes"
...A chapter on dioids can be found in Gondran et Minoux [21]....
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...A complete account of existing algorithms can be found in [21]....
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...For the homogeneous system of n linear equations with n unknowns, Axr0 has a signed non zero solution iff detAr0 (see [34,18]), which extends a result of Gondran and Minoux (see [22])....
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...The theory of linear independence using bideterminants, which is the ancester of symmetrization, was initiated by Gondran and Minoux (following Kuntzmann)....
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