On the Freezing of Variables in Random Constraint Satisfaction Problems

Guilhem Semerjian

doi:10.1007/s10955-007-9417-7

Journal of Statistical Physics

January 2008, Volume 130, Issue 2, pp 251–293

On the Freezing of Variables in Random Constraint Satisfaction Problems

Authors
Authors and affiliations

Guilhem SemerjianEmail author

Guilhem Semerjian
- 1
Email author

1.LPTENS, Unité Mixte de Recherche (UMR 8549) du CNRS et de l’ENS associée à l’Université Pierre et Marie CurieParis Cedex 05France

Article

First Online:: 03 October 2007

Received:: 16 May 2007
Accepted:: 27 August 2007

DOI: 10.1007/s10955-007-9417-7

Cite this article as:: Semerjian, G. J Stat Phys (2008) 130: 251. doi:10.1007/s10955-007-9417-7

35 Citations
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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.

Keywords

Random constraint satisfaction problemsFreezing transitionCavity method

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References

1.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1983) Google Scholar
2.
Schaefer, T.: In: Proceedings of the 10th Annual ACM Symposium on the Theory of Computing, San Diego, p. 216 (1978)
3.
Janson, S., Luczak, T., Rucinski, A.: Random Graphs. Wiley, New York (2000) MATHGoogle Scholar
4.
Mitchell, D., Selman, B., Levesque, H.: In: Proc. of the 10th National Conference on Artificial Intelligence, p. 459 (1992)
5.
Mézard, M., Ricci-Tersenghi, F., Zecchina, R.: J. Stat. Phys. 111, 505 (2003) MATHCrossRefGoogle Scholar
6.
Cocco, S., Dubois, O., Mandler, J., Monasson, R.: Phys. Rev. Lett. 90, 047205 (2003) CrossRefADSGoogle Scholar
7.
Goerdt, A.: J. Comput. Sci. 53, 469 (1996) MATHCrossRefMathSciNetGoogle Scholar
8.
Friedgut, E.: J. Am. Math. Soc. 12, 1017 (1999) MATHCrossRefMathSciNetGoogle Scholar
9.
Franco, J.: Theor. Comput. Sci. 265, 147 (2001) MATHCrossRefMathSciNetGoogle Scholar
10.
Dubois, O.: Theor. Comput. Sci. 265, 187 (2001) MATHCrossRefGoogle Scholar
11.
Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987) MATHGoogle Scholar
12.
Monasson, R., Zecchina, R.: Phys. Rev. Lett. 76, 3881 (1996) MATHCrossRefADSMathSciNetGoogle Scholar
13.
van Mourik, J., Saad, D.: Phys. Rev. E 66, 056120 (2002) CrossRefADSGoogle Scholar
14.
Biroli, G., Monasson, R., Weigt, M.: Eur. Phys. J. B 14, 551 (2000) CrossRefADSGoogle Scholar
15.
Mézard, M., Parisi, G., Zecchina, R.: Science 297, 812 (2002) CrossRefADSGoogle Scholar
16.
Mézard, M., Zecchina, R.: Phys. Rev. E 66, 056126 (2002) CrossRefADSGoogle Scholar
17.
Maneva, E.N., Mossel, E., Wainwright, M.J.: Proceedings of the Symposium on Discrete Algorithms, Vancouver, January 2005
18.
Mézard, M., Mora, T., Zecchina, R.: Phys. Rev. Lett. 94, 197205 (2005) CrossRefADSGoogle Scholar
19.
Achlioptas, D., Ricci-Tersenghi, F.: arXiv:cs/0611052. In: Proceedings of the Symposium on the Theory of Computing (2006)
20.
Gopalan, P., Kolaitis, P., Maneva, E., Papadimitriou, C.H.: In: Proceedings of ICALP 2006, p. 346 (2006)
21.
Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborová, L.: Proc. Nat. Acad. Sci. 104, 10318 (2007) CrossRefMATHADSGoogle Scholar
22.
Krzakala, F., Zdeborová, L.: arXiv:0704.1269 (2007)
23.
Montanari, A., Ricci-Tersenghi, F., Semerjian, G.: In preparation
24.
Mertens, S., Mézard, M., Zecchina, R.: Random Struct. Alg. 28, 340 (2006) MATHCrossRefGoogle Scholar
25.
Krzakala, F., Pagnani, A., Weigt, M.: Phys. Rev. E 70, 046705 (2004) CrossRefADSGoogle Scholar
26.
Mézard, M., Palassini, M., Rivoire, O.: Phys. Rev. Lett. 95, 200202 (2005) CrossRefGoogle Scholar
27.
Montanari, A., Semerjian, G.: J. Stat. Phys. 124, 103 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
28.
Zhou, H.: New J. Phys. 7, 123 (2005) CrossRefADSGoogle Scholar
29.
Kschischang, F.R., Frey, B.J., Loeliger, H.-A.: IEEE Trans. Inform. Theory 47, 498 (2001) MATHCrossRefMathSciNetGoogle Scholar
30.
Mézard, M., Parisi, G.: Eur. Phys. J. B 20, 217 (2001) CrossRefADSGoogle Scholar
31.
Lindvall, T.: Lectures on the Coupling Method. Dover, New York (2002) MATHGoogle Scholar
32.
Götze, W., Sjögren, L.: Rep. Prog. Phys. 55, 241 (1992) CrossRefGoogle Scholar
33.
Mossel, E.: In: Nestril, J., Winkler, P. (eds.) Graphs, Morphisms and Statistical Physics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence (2004) Google Scholar
34.
Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, Berlin (1988) MATHGoogle Scholar
35.
Mézard, M., Montanari, A.: J. Stat. Phys. 124, 1317 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
36.
Talagrand, M.: Spin Glasses: a Challenge for Mathematicians. Springer, Berlin (2003) Google Scholar
37.
Gerschenfeld, A., Montanari, A.: arXiv:0704.3293 (2007)
38.
Guerra, F.: Comm. Math. Phys. 233, 1 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
39.
Franz, S., Leone, M.: J. Stat. Phys. 111, 535 (2003) MATHCrossRefMathSciNetGoogle Scholar
40.
Panchenko, D., Talagrand, M.: Probab. Theory Relat. Fields 130, 319 (2004) MATHCrossRefMathSciNetGoogle Scholar
41.
Talagrand, M.: Ann. Math. 163, 221 (2006) MATHMathSciNetCrossRefGoogle Scholar
42.
Montanari, A., Parisi, G., Ricci-Tersenghi, F.: J. Phys. A 37, 2073 (2004) MATHCrossRefADSMathSciNetGoogle Scholar
43.
Aldous, D., Steele, J.M.: In: Kesten, H. (ed.) Probability on Discrete Structures, p. 1. Springer, Berlin (2003) Google Scholar
44.
Aldous, D., Bandyopadhyay, A.: Ann. Appl. Probab. 15, 1047 (2005) MATHCrossRefMathSciNetGoogle Scholar
45.
Bandyopadhyay, A., Gamarnik, D.: In: Random Structures and Algorithms. math.PR/0510471 (2005, to appear) (preliminary version in Proceedings of SODA 2006)
46.
Montanari, A., Shah, D.: In: Proceedings of SODA 2007. cs.DM/0607073 (2006)
47.
Bayati, M., Nair, C.: In: Annual Allerton Conference on Communication, Control and Computing. cond-mat/0607290 (2006)
48.
Montanari, A., Ricci-Tersenghi, F.: Eur. Phys. J. B 33, 339 (2003) CrossRefADSGoogle Scholar
49.
Mora, T., Mézard, M.: J. Stat. Mech., P10007 (2006)
50.
Duxbury, P.M., Jacobs, D.J., Thorpe, M.F., Moukarzel, C.: Phys. Rev. E 59, 2084 (1999) CrossRefADSGoogle Scholar
51.
Schwarz, J.M., Liu, A.J., Chayes, L.Q.: Europhys. Lett. 73, 560 (2006) CrossRefADSGoogle Scholar
52.
Sellitto, M., Biroli, G., Toninelli, C.: Europhys. Lett. 69, 496 (2005) CrossRefADSGoogle Scholar
53.
Montanari, A., Semerjian, G.: Unpublished (2005)
54.
Krzakala, F., Kurchan, J.: cond-mat/0702546 (2007)
55.
Castellani, T., Napolano, V., Ricci-Tersenghi, F., Zecchina, R.: J. Phys. A 36, 11037 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
56.
Flajolet, P., Odlyzko, A.: SIAM J. Discrete Math. 3, 216 (1990) MATHCrossRefMathSciNetGoogle Scholar

On the Freezing of Variables in Random Constraint Satisfaction Problems

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