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Solutions optimales des problèmes de recouvrement sous contraintes sur le degré des nœuds

Abstract : The work conducted in this thesis is focused on the minimum spanning problems in graphs under constraints on the vertex degrees. As the spanning tree covers the vertices of a connected graph with a minimum number of links, it is generally proposed as a solution for this kind of problems. However, for some applications such as the routing in optical networks, the solution is not necessarily a sub-graph. In this thesis, we assume that the degree constraints are due to a limited instantaneous capacity of the vertices and that the only pertinent requirement on the spanning structure is its connectivity. In that case, the solution may be different from a tree. We propose the reformulation of this kind of spanning problems. To find the optimal coverage of the vertices, an extension of the tree concept called hierarchy is proposed. Our main purpose is to show its interest regarding the tree in term of feasibility and costs of the coverage. Thus, we take into account two types of degree constraints: either an upper bound on the degree of vertices and an upper bound on the number of branching vertices. We search a minimum cost spanning hierarchy in both cases. Besides, we also illustrate the applicability of hierarchies by studying a problem that takes more into account the reality of the optical routing. For all those NP-hard problems, we show the interest of the spanning hierarchy for both costs of optimal solutions and performance guarantee of approximate solutions. These results are confirmed by several experimentations on random graphs.
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Massinissa Merabet. Solutions optimales des problèmes de recouvrement sous contraintes sur le degré des nœuds. Algorithme et structure de données [cs.DS]. Université Montpellier II - Sciences et Techniques du Languedoc, 2014. Français. ⟨NNT : 2014MON20138⟩. ⟨tel-01684758⟩

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