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On the Freezing of Variables in Random Constraint Satisfaction Problems

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Abstract

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we introduce and study a message passing procedure that allows to compute, for generic constraint satisfaction problems, the sizes of the rearrangements induced in response to the modification of a variable. These sizes diverge at the freezing transition, with a critical behavior which is also investigated in details. We apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.

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  1. LPTENS, Unité Mixte de Recherche (UMR 8549) du CNRS et de l’ENS associée à l’Université Pierre et Marie Curie, 24 Rue Lhomond, 75231, Paris Cedex 05, France

    Guilhem Semerjian

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  1. Guilhem Semerjian
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Semerjian, G. On the Freezing of Variables in Random Constraint Satisfaction Problems. J Stat Phys 130, 251–293 (2008). https://doi.org/10.1007/s10955-007-9417-7

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  • DOI: https://doi.org/10.1007/s10955-007-9417-7

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