# Multiflots, métriques et graphes h-parfaits : les cycles impairs dans l'optimisation combinatoire

Abstract : This work deals with problems in Combinatorial Optimization. We study in particular some objects for which problems, that usually are NP-complete, turn out to be polynomial. First, we consider the problem of the multiflow feasibility, that has many applications in Operations Research. Given the specification of one such problem, with the network, the capacities and demands, we look for a proof of existence or non-existence of a solution. One way to treat this problem is to give necessary and sufficient conditions for the existence of a multiflow, as the one known as cut condition''. We introduce the (CC, K_5, F_7) condition, that generalizes the cut condition and specialize the existent CC3 condition. The structure of the multiflow problem led us to consider a related problem, namely the metric packing. We treat the integer and the half-integer packing concerning the CC3, K_5 and F_7 metrics. We characterize the class of graphs and, more generally, the class of matroids, for which there exist integer and half-integer packings, under some additional hypotheses. Then, we study general properties of h- and t-perfect graphs, and the associated coloring problem. We present some bounds for the chromatic number, and classes of h- and t-perfect graphs that satisfy a conjecture of Shepherd. Finally, we show the hierarchy of the graphs studied in this document, that is obtained using tools as weakly bipartite graphs, binary clutters and 0-1 matrices. We close pointing out some directions of research that arise with this work, both on the feasibility of multiflows and on the coloring of h- and t-perfect graphs.
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Submitted on : Monday, February 23, 2004 - 4:14:17 PM
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• HAL Id : tel-00005002, version 1

### Citation

Karina Marcus. Multiflots, métriques et graphes h-parfaits : les cycles impairs dans l'optimisation combinatoire. Modélisation et simulation. Université Joseph-Fourier - Grenoble I, 1996. Français. ⟨tel-00005002⟩

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