Landscape analysis of constraint satisfaction problems

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 76, Issue 2, 2007, Pages

  Krzakala, Florent ^a   Kurchan, Jorge ^b  

^a Centre National de la Recherche Scientifique   (France)

^b ESPCI   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BENCHMARKING; GRAPH THEORY; PROBLEM SOLVING; STATISTICAL METHODS; THERMODYNAMICS;

LANDSCAPE ANALYSIS; PACKING PROBLEMS; RANDOM CLOSE PACKING; SPHERE PACKING;

CONSTRAINT THEORY;

EID: 34548449427     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.76.021122     Document Type: Article

References (72)

1. M. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory and NP-Completeness (Freeman, San Francisco, 1979);
2. C. H. Papadimitriou, Computational Complexity (Addison-Wesley, New York, 1994).
3. M. Mézard and R. Zecchina, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.056126 66, 056126 (2002);
4. M. Mézard, G. Parisi, and R. Zecchina, Science SCIEAS 0036-8075 10.1126/science.1073287 297, 812 (2002).
5. R. Monasson and R. Zecchina, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.56.1357 56, 1357 (1997).
6. R. Mulet, A. Pagnani, M. Weigt, and R. Zecchina, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.89.268701 89, 268701 (2002).
7. Y. Fu and P. W. Anderson, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/19/9/033 19, 1605 (1986).
8. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
9. F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborová, Proc. Natl. Acad. Sci. U.S.A. PNASA6 0027-8424 10.1073/pnas.0703685104 104, 10318 (2007).
10. C. S. O'Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.68.011306 68, 011306 (2003).
11. For compression procedure closely related to the experimental one, and yielding densitites close to random close packing, see Hernán A. Makse, David L. Johnson, and Lawrence M. Schwartz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.4160 84, 4160 (2000).
12. B. D. Lubachevsky and F. H. Stillinger, J. Stat. Phys. JSTPBS 0022-4715 10.1007/BF01025983 60, 561 (1990).
13. F. H. Stillinger and T. A. Weber, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.449840 83, 4767 (1985).
14. A.-S. Sznitman, Brownian Motion, Obstacles and Random Media (Springer, Berlin, 1998).
15. This is perhaps related to the formalism in P. M. Duxbury, D. J. Jacobs, M. F. Thorpe, and C. Moukarzel, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.59.2084 59, 2084 (1999).
16. A potential is also defined in C. Brito and M. Wyart, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2006-10412-2 76, 986 (2006).
17. C. De Dominicis, M. Gabay, T. Garel, and H. Orland, J. Phys. (France) JOPQAG 0302-0738 41, 922 (1980).
18. A. J. Bray and M. A. Moore, J. Phys. C JPSOAW 0022-3719 10.1088/0022-3719/13/19/002 13, L469 (1980).
19. L. F. Cugliandolo and J. Kurchan, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/27/17/011 27, 5749 (1994).
20. The literature on the metastable states of the Sherrington-Kirkpatrick model covers a quarter of a century. For recent works, see A. Crisanti, L. Leuzzi, G. Parisi, and T. Rizzo, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.70.064423 70, 064423 (2004);
21. M. Muller, L. Leuzzi, and A. Crisanti, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.74.134431 74, 134431 (2006);
22. A. Cavagna, I. Giardina, G. Parisi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.120603 92, 120603 (2004).
23. The barrier to escape a state of free energy per spin f scales as B (f- f0) -1? 3, where f0 is the free energy density of the lowest state; a finite number of flips will unstabilize states with f> f0, see T. Aspelmeier, R. A. Blythe, A. J. Bray, and M. A. Moore, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.74.184411 74, 184411 (2006). Thus, these calculations are in agreement with the conclusions from the dynamics.
24. J. Kurchan, G. Parisi, and M. A. Virasoro, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1993217 3, 1819 (1993).
25. A. Crisanti and H.-J. Sommers, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1995164 5, 805 (1995).
26. J. Kurchan and L. Laloux, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/29/9/009 29, 1929 (1996).
27. L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.71.173 71, 173 (1993).
28. L. F. Cugliandolo, D. Grempel, G. Lozano, and H. Lozza (unpublished).
29. L. Zdeborová and F. Krzakala, e-print arXiv:0704.1269v1.
30. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian (unpublished).
31. A. Montanari and F. Ricci-Tersenghi, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2003-00174-7 33, 339 (2003).
32. A. Montanari and F. Ricci-Tersenghi, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.70.134406 70, 134406 (2004);
33. B. Capone, T. Castellani, I. Giardina, and F. Ricci-Tersenghi, Physica B PHYBE3 0921-4526 74, 144301 (2006);
34. A. Barrat, S. Franz, and G. Parisi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/30/16/006 30, 5593 (1997).
35. Andrea Montanari, Giorgio Parisi, and Federico Ricci-Tersenghi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/37/6/008 37, 2073 (2004).
36. Florent Krzakala, Andrea Pagnani, and M. Weigt Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.046705 70, 046705 (2004).
37. R. Monasson and D. O'Kane, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/27/2/002 27, 85 (1994);
38. G. Biroli, R. Monasson, and M. Weigt, Eur. Phys. J. B EPJBFY 1434-6028 10.1007/s100510051065 14, 551 (2000);
39. M. Mézard, M. Palassini, and O. Rivoire, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.200202 95, 200202 (2005);
40. also see Ref..
41. A. Barrat, e-print arXiv:cond-mat/9701031.
42. J. van Mourik and D. Saad, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.056120 66, 056120 (2002).
43. S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Nature (London) NATUAS 0028-0836 10.1038/31146 393, 554 (1998).
44. B. Selman, H. A. Kautz, and B. Cohen, Proceedings of the AAAI-94, Seattle, 1994.
45. John Ardelius and Erik Aurell, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.037702 74, 037702 (2006).
46. J. Whittemore, J. Kim, and K. Sakallah "SATIRE: a new incremental satisfiability engine," in Proceedings of the 38th conference on design automation, 2001, pp. 542-545;
47. N. En and N. Srensson, "Temporal induction by incremental SAT solving," First international workshop on bounded model checking, ENTCS, Vol. 89 (4).
48. A. Montanari and G. Semerjian, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.247201 94, 247201 (2005);
49. A. Montanari and G. Semerjian, J. Stat. Phys. JSTPBS 0022-4715 124, 1572 (2006).
50. In fact the model considered in is somewhat special. More general and detailed results are obtained for the coloring and the satisfiability problem in G. Semerjian, e-print arXiv:0705.2147.
51. D. Achlioptas and F. Ricci-Tersenghi, Report No. CC/0611052.
52. For an overview, see M. Wyart, Ann. Phys. (Paris) ANPHAJ 0003-4169 10.1051/anphys:2006001 30, 1 (2005).
53. Reference was criticized in A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.043301 70, 043301 (2004).
54. See the reply: C. S. O'Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.043302 70, 043302 (2004).
55. M. Wyart, S. R. Nagel, and T. A. Witten, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2005-10245-5 72, 486 (2005).
56. S. Torquato, T. M. Truskett, and P. G. Debenedetti, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.2064 84, 2064 (2000).
57. M. Skoge, A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.041127 74, 041127 (2006).
58. W. Kob, F. Sciortino, and P. Tartaglia, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2000-00191-8 49, 590 (2000).
59. L. F. Cugliandolo, e-print arXiv:cond-mat/0210312.
60. G. Parisi and M. A. Viarsoro, J. Phys. (Paris) JOPQAG 0302-0738 50, 3317 (1989).
61. R. Monasson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.75. 2847 75, 2847 (1995).
62. S. Franz and G. Parisi, Physica A PHYADX 0378-4371 10.1016/S0378-4371(98) 00315-X 261, 317 (1998).
63. S. Cocco and R. Monasson, Theor. Comput. Sci. TCSCDI 0304-3975 10.1016/j.tcs.2004.02.034 320, 345 (2004).
64. The clustering point is the celebrated mode-coupling (dynamic) transition point of glass theory. This has been recently extended to dilute systems in Andrea Montanari and Guilhem Semerjian, J. Stat. Phys. JSTPBS 0022-4715 10.1007/s10955-006-9175-y 125, 23 (2006).
65. A. J. Liu, and S. R. Nagel, Nature (London) NATUAS 0028-0836 396, 6706 (1998).
66. See the discussion for hard spheres in F. Zamponi, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.75.043101 75, 043101 (2007).
67. J. M. Schwarz, A. J. Liu, and L. Q. Chayes, e-print arXiv:cond-mat/ 0410595.
68. C. F. Moukarzel, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett. 81.1634 81, 1634 (1998).
69. An additional puzzle is brought about by the replica calculations, where states are isostatic but not critical, see G. Parisi and F. Zamponi, J. Stat. Mech.: Theory Exp. ZZZZZZ 1742-5468 (2006) 3017;
70. G. Parisi and F. Zamponi, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.2041507 123, 144501 (2005).
71. P. M. Chaikin, A. Donev, W. N. Man, F. H. Stillinger, and S. Torquato, Ind. Eng. Chem. Res. IECRED 0888-5885 10.1021/ie060032g 45, 6960 (2006);
72. A. Donev, R. Connely, F. H. Stillinger, and S. Torquato (unpublished).