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Phases vitreuses, optimisation et grandes déviations

Abstract : Combinatorial optimization problems lie at the core of the theory of algorithmic complexity. They are also tightly related to a mean-field formulation of lattice models of spin glasses and structural glasses, known as the Bethe approximation. This thesis relies on this parallel to apply to optimization problems an approach originating from the statistical physics of disordered systems, the cavity method. Given an ensemble of instances of an optimization problem, this method allows one to determine the properties of solutions of typical instances, as well as those of atypical instances whose probabilities are exponentially small (large deviations on the external structure). For a given instance, the cavity method also gives access to the thermodynamics of the different admissible solutions (large deviations on the internal structure). From a physical perspective, many algorithmically hard optimization problems are thus found to display a glassy-like phase. This thesis is composed of three parts intended to present the principles, applications and limitations of the cavity method. The first part recalls, from the point of view of large deviations, the links between statistical physics and combinatorial optimization. The second part treats models defined on random graphs and, for different ensembles of graphs, analyzes the typical and atypical properties of these models. The third part is devoted to large deviations on the "internal disorder", constituted by the solutions and quasi-solutions of a given instance. The emphasis is put on glassy phases, where the set of solutions is broken into an exponentially large number of disjoint clusters (the so-called one-step replica symmetry breaking structure); it is shown how the cavity method yields in such cases a detailed characterization of the geometrical properties of the space of solutions.
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Contributor : Olivier Rivoire Connect in order to contact the contributor
Submitted on : Monday, August 22, 2005 - 12:04:12 PM
Last modification on : Wednesday, October 6, 2021 - 3:51:24 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:09:39 PM


  • HAL Id : tel-00009956, version 1



Olivier Rivoire. Phases vitreuses, optimisation et grandes déviations. Analyse de données, Statistiques et Probabilités []. Université Paris Sud - Paris XI, 2005. Français. ⟨tel-00009956⟩



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