Simplest random K-satisfiability problem

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

Volumn 63, Issue 2 II, 2001, Pages 1-11

  Ricci Tersenghi, Federico ^a   Weigt, Martin ^b   Zecchina, Riccardo ^a  

^a ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS   (Italy)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18344377121     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.026702     Document Type: Article

References (41)

1. M. Garey and D.S. Johnson, Computers and Intractability; A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979); C. Papadimitriou, Computational Complexity (Addison-Wesley, Reading, MA 1994).
2. M. Garey and D.S. Johnson, Computers and Intractability; A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979); C. Papadimitriou, Computational Complexity (Addison-Wesley, Reading, MA 1994).
3. S.A. Cook, in Proceedings of the Third Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1971), p. 151.
4. Y.-T. Fu and P.W. Anderson, in Lectures in the Sciences of Complexity, edited by D. Stein (Addison-Wesley, Reading, MA, 1989), p. 815.
5. D. Mitchell, B. Selman, and H. Levesque, in Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92) (AAAI Press, Menlo Park, CA, 1992) p. 459.
6. S. Kirkpatrick and B. Selman, Science 264, 1297 (1994).
7. Artif. Intel. 81, issue 1-2 (1996).
8. R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky, Nature (London) 400, 133 (1999).
9. M. Weigt and A.K. Hartmann, Phys. Rev. Lett. 84, 6118 (2000).
10. D. Achlioptas, C. Gomes, D. Liang, H. Kautz, and B. Selman (unpublished).
11. B. Bollobás, C. Borgs, J.T. Chayes, J.H. Kim, and D.B. Wilson, e-print arXiv:math.CO/9909031.
12. c(K).
13. A. Goerdt, in Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science in Lecture Notes in Computer Science Vol. 629 (Springer, Berlin, 1992), p. 264; J. Comput. Syst. Sci. 53, 469 (1996); V. Chvàtal and B. Reed, in Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (IEEE, New York, 1992), p. 620.
14. A. Goerdt, in Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science in Lecture Notes in Computer Science Vol. 629 (Springer, Berlin, 1992), p. 264; J. Comput. Syst. Sci. 53, 469 (1996); V. Chvàtal and B. Reed, in Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (IEEE, New York, 1992), p. 620.
15. A. Goerdt, in Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science in Lecture Notes in Computer Science Vol. 629 (Springer, Berlin, 1992), p. 264; J. Comput. Syst. Sci. 53, 469 (1996); V. Chvàtal and B. Reed, in Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (IEEE, New York, 1992), p. 620.
16. B. Aspvall, M.F. Plass, and R.E. Tarjan, Inf. Process. Lett. 8, 121 (1979).
17. M. Mézard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
18. D. Gross and M. Mézard, Nucl. Phys. B 240, 431 (1984).
19. E. Gardner, Nucl. Phys. B 257, 747 (1985).
20. E. Gardner and B. Derrida, J. Phys. A 22, 1983 (1989).
21. T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. Lett. 58, 2091 (1987).
22. A. Barrat and R. Zecchina, Phys. Rev. E 59, 1299 (1999).
23. i) appears in the ℓth clause, and 0 otherwise. Comparison of the above expression with Eq. (3) clarifies the difference with hSAT in which only the 3-spin interaction term appears.
24. D. Achlioptas (unpublished).
25. B. Selman, H.A. Kautz and B. Cohen, Cliques, Coloring, and Satisfiability, edited by D.S. Johnson and M.A. Trick, DI-MACS Series in Discrete Mathematics and Theoretical Computer Science Vol. 26 (AMS, Providence, RI, 1996).
26. J.M. Crawford and L.D. Auton, in Proceedings of the Eleventh National Conference on Artificial Intelligence (AAAI-93) (AAI Press, Menlo Park, CA, 1993), p. 21.
27. The Boolean expressions which are generated in this way are simply those which have the highest probability given the satisfying assignment and therefore are those with maximum entropy, i.e., the easiest to satisfy.
28. G. Biroli, R. Monasson, and M. Weigt, Eur. Phys. J. B 14, 551 (2000).
29. However, the entropy is not trivially given by ln(2) times the fraction of variables which do not belong to the backbone.
30. M. Davis and H. Putnam, J. Assoc. Comput. Mach. 7, 201 (1960).
31. R. Monasson and R. Zecchina, Phys. Rev. Lett. 76, 3881 (1996); Phys. Rev. E 56, 1357 (1997).
32. R. Monasson and R. Zecchina, Phys. Rev. Lett. 76, 3881 (1996); Phys. Rev. E 56, 1357 (1997).
33. B. Bollobas, Random Graphs (Academic Press, London, 1985).
34. Note that in random hypergraphs the probability that any pair of vertices belongs to more that one hyperlink is proportional to 1/N for large N, we have not considered this possibility when constructing the structures in Fig. 1.
35. ℓ.
36. M.-T. Chao and J. Franco, Inf. Sci. (N.Y.) 51, 289 (1990).
37. D.B. Wilson, e-print arXiv:math/0005136.
38. c of the curves in Fig. 2. Other methods give compatible results.
39. J. Berg, S. Franz, M. Leone, F. Ricci-Tersenghi, and R. Zecchina (unpublished).
40. In any SAT phase proving satisfiability is equivalent to finding a ground state whereas in an UNSAT phase the proof of unsatisfiability does not provide any information on the precise assignments of the variables in the ground states.
41. Similarly to what happens in Max-KSAT. See, for instance, P. Beame, R. Karp, T. Pitassi, and M. Saks, in Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing (ACM, New York, 1998), p. 561.