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On the design of networks with unicyclic connected components Makhlouf Hadji · Walid Ben-Ameur
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Abstract In this thesis, we use the polyhedral approach to solve combinatorial problems in telecommunications context. First, we introduce the problem of network design with unicyclic connected components. We recall that without other constraints, our problem is easy to solve, and we propose a study with new technical constraints. We start our study by adding constraints on the size of cycles. We aim to obtain unicyclic components such that the size of each cycle is not lower than a certain p. This problem is NP-Hard. We describe some valid inequalities for the design of unicyclic graphs with girth constraints. The faces induced by these valid inequalities are also studied. Some of them can be separated in polynomial time. A cutting plane algorithm based on these inequalities is implemented to solve the problem. Furthermore, we focus on a Steiner type problem, which consists in partitioning the graph to unicyclic components, such that some given vertices belong to a cycle. We prove then that our problem is easy to solve, and we propose an exact extended formulation and a partial description of the convex hull of the incidence vectors of our Steiner network problem. Polynomial time separation algorithms are described. One of them is a generalization of the M. Hadji 09, Rue Charles Fourier, 91010 Evry, France Tel.: +336-79-412728 Fax: +331-60-764578 E-mail:
[email protected] W. Ben-Ameur 09, Rue Charles Fourier, 91010 Evry, France Tel.: +331-60-764713 Fax: +331-60-764578 E-mail:
[email protected]
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Makhlouf Hadji, Walid Ben-Ameur
Padberg&Rao algorithm to separate blossom inequalities. Other technical constraints are proposed such as degree constraints, a bound of the number of unicyclic components, constraints related to whether some given pairs of vertices belong to the same component or to different components. Finally, we study the spectra of two specified classes of unicyclic graphs.
