# Survivale Network Design Problems with High Connectivity Requirement

Abstract : This thesis is part of a polyhedral study design problems of reliable networks with high connectivity. In particular, we consider the problem of subgraph called k-edge-connected network design and k-edge-connected with terminal constraint when k _> 3. In a first part, we study the problem of sub-graph k-edge-connected. Given an undirected graph and valued G = (V, E) and a positive integer k, the problem of subgraph k-edge-connected is to determine a subgraph of G of minimum weight k such that there edge-disjoint chains between each pair of vertices in V. We discuss the polytope associated with that problem when k _> 3. We introduce a new family of valid inequalities for the polytope and present several families of valid inequalities. For each family of inequalities, we investigate the conditions under which these inequalities define facets. We also discuss the problem of separation associated with each family of inequalities and reduction operations graphs. Using these results, we develop an algorithm cuts and connections for the problem and give experimental results. Then, we study the problem of network design k-edge-connected with terminal constraint. Let G = (V, E) be a weighted graph undirected, a set of demands D _C V x V and two positive integers k and L. The problem of network design k-edge-connected with terminal constraint is to determine a subgraph of G of minimum weight such that between each pair of vertices {s, t} ED, there are k edge-disjoint chains of length at most L. We study this problem in the case where k _> 2 and THE {2, 3}. We examine the structure of the associated polytope and show that, when IDI = 1, this polytope is completely described by the inequalities called st-cut and L-path-cut inequalities with trivial. This result generalizes those of Huygens et al. [75] for k = 2, L E {2, 3} and Dahl et al. [35] for k _> 2, L = 2. Finally, we consider the problem of network design k-edge-connected with terminal constraint when k _> 2, THE {2, 3} and IDI _> 2. The problem is NP-hard in this case. We introduce four new formulations of the problem in the form of integer linear programs. These are based on the transformation of the graph G suitable directed graphs. We discuss the polytope associated with each formulation and introduce several families of valid inequalities. For each, we describe conditions for these inequalities define facets. Using these results, we develop algorithms and cuts and cuts connections, connections and column generation for the problem. We give experimental results and conduct a comparative study between the different formulations.
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https://tel.archives-ouvertes.fr/tel-00724580
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Submitted on : Tuesday, August 21, 2012 - 4:19:08 PM
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• HAL Id : tel-00724580, version 1

### Citation

Ibrahima Diarrassouba. Survivale Network Design Problems with High Connectivity Requirement. Networking and Internet Architecture [cs.NI]. Université Blaise Pascal - Clermont-Ferrand II, 2009. English. ⟨NNT : 2009CLF21989⟩. ⟨tel-00724580⟩

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