Réseaux d'interconnexion bipartis : colorations généralisées dans les graphes

Étude sur les graphes bipartis orientes de Moore montrant que de tels graphes existent, pour certaines valeurs du diamètre, et servent a la construction d'une classe de graphes bipartis orientes, asymptotiquement optimaux. Dans la deuxième partie du travail, quelques notions de coloration des graphes sont présentées. Celles-ci permettent de généraliser certains résultats déjà connus dans le cadre de la coloration habituelle et d'en obtenir d'autres plutôt spécifiques a ces notions. La généralisation de la notion de perfection en b-perfection est proposée ce qui permet l'obtention des graphes triangules représentant la seule classe de graphes b-parfait

Relaxed Locally Identifying Coloring of Graphs

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Solving a dynamic combinatorial auctions problem by a hybrid metaheuristic based on a fuzzy dominance relation

International audienceThis paper introduces a bi-objective winner determination problem which is based on English auctions. Most models of combinatorial auctions (winner determination problem) do not allow the bidder to update his offer, due to the fact that these mechanisms are static. However in reality bidders are in rough competition while there is time for auction. In this work we give a mathematical formulation of the dynamic model of the bi-objective winner determination problem, where the objectives are: (i) maximization of the total income, (ii) maximization of the number of items sold. This problem is based on the English auction mechanism, which allows bidders to renew their bids until the end of the exercise period. Then the solution is proposed by giving an algorithm based on an hybridization of a metaheuristic with a fuzzy dominance relation. A numerical experimentation using this algorithm on simulated data gives rise to satisfactory results

On the star forest polytope for trees and cycles

International audienceLet G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is N P-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SF P (G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SF P (G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SF P (G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SF P (G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652-661], we give a complete linear description of SF P (G) when G is a cycle