Random MAX SAT, random MAX CUT, and their phase transitions

Random Structures and Algorithms

Volumn 24, Issue 4, 2004, Pages 502-545

  Coppersmith, Don ^a   Gamarnik, David ^a   Hajiaghayi, MohammadTaghi ^b   Sorkin, Gregory B ^a  

^a IBM T J WATSON RESEARCH CENTER   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 6344228835     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/rsa.20015     Document Type: Article

References (50)

1. D. Achlioptas and C. Moore, The asymptotic order of the &-SAT threshold, Proc 43rd Annu IEEE Symp Found Comput Sci, 2002, pp. 779-788.
2. 0(1)), Proc 35th Annu ACM Symp Theory Computing, 2003, pp. 223-231.
3. D. Achlioptas and G. B. Sorkin, Optimal myopic algorithms for random 3-SAT, 41st Annu Symp Found Comput Sci, IEEE Comput Soc Press, Los Alamitos, CA, 2000, pp. 590-600.
4. D. Achlioptas, A. Naor, and Y. Peres, On the maximum satisfiability of random formulas, Proc 44th Symp Found Comput Sci (FOCS 2003), IEEE Computer Society, October 2003, pp. 362-370.
5. D. Aldous, Asymptotics in the random assignment problem, Probab Theory Related Fields 93(4) (1992), 507-534, MR 94b:60013.
6. D. J. Aldous, The ζ(2) limit in the random assignment problem, Random Structures Algorithms 18(4) (2001), 381-418, MR 2002f:60015.
7. D. Aldous and J. M. Steele, The objective method: Probabilistic combinatorial optimization and local weak convergence, preprint, 2002.
8. A. Bertoni, P. Campadelli, and R. Posenato, An upper bound for the maximum cut mean value, Graph-Theoretic Concepts in Computer Science, Berlin, 1997, Lecture Notes in Computer Science 1335, Springer-Verlag, Berlin, 1997, pp. 78-84, MR 99d:68185.
9. T. Bohman and A. M. Frieze, Avoiding a giant component, Random Structures Algorithms 19(1) (2001), 75-85.
10. T. Bohman, A. M. Frieze, and N. C. Wormald, Avoiding a giant component II, unpublished manuscript, February 2002.
11. B. Bollobás, "Martingales, isoperimetric inequalities and random graphs," Combinatorics, Eds. A. Hajnal, L. Lovász, and V. T. Sós, Colloq Math Soc János Bolyai, No. 52, North Holland, Amsterdam, 1988, pp. 113-139.
12. B. Bollobás, Modern graph theory, Springer-Verlag, New York, 1998.
13. B. Bollobás, C. Borgs, J. T. Chayes, J. H. Kim, and D. B. Wilson, The scaling window of the 2-SAT transition, Random Structures Algorithms 18(3) (2001), 201-256.
14. A. Z. Broder, A. M. Frieze, and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proc Fourth Annu ACM-SIAM Symp Discrete Algorithms, Austin, TX, 1993, ACM, New York, 1993, pp. 322-330, MR 94b:03023.
15. V. Chvátal and B. Reed, Mick gets some (the odds are on his side), 33th Annu Symp Found Comput Sci, Pittsburgh, PA, IEEE Comput Soc Press, Los Alamitos, CA, 1992, pp. 620-627.
16. A. Coja-Oghlan, C. Moore, and V. Sanwalani, MAX k-CUT and approximating the chromatic number of random graphs, Proc Thirtieth Int Coll Automata, Languages, Programming, Lecture Notes in Computer Science 2719, Springer, 2003, pp. 200-211.
17. D. Coppersmith, D. Gamarnik, M. Hajiaghayi, and G. B. Sorkin, Random MAX SAT, random MAX CUT, and their phase transitions, Proc 14th Annu ACM-SIAM Symp Discrete Algorithms, Baltimore, MD, 2003, ACM, New York, January 2003, pp. 364-373.
18. N. Creignou and H. Daude, Satisfiability threshold for random XOR-CNF formulas, Discrete Appl Math 96/97 (1999), 41-53, The satisfiability problem, Certosa di Pontignano, 1996; Boolean functions, MR 20001:68072.
19. N. Creignou and H. Daudé, Smooth and sharp thresholds for random k-XOR-CNF satisfiability, Theor Inform Appl 37(2) (2003), 127-147, MR 2 015 688.
20. N. Creignou, H. Daudé, and O. Dubois, Approximating the satisfiability threshold for random k-XOR-formulas, Combin Probab Comput 12(2) (2003), 113-126, MR 2004b:68119.
21. J. Díaz, N. Do, M. J. Serna, and N. C. Wormald, Bounds on the max and min bisection of random cubic and random 4-regular graphs, Theoret Comput Sci 307(3) (2003), 531-547, Selected papers in honor of Lawrence Harper, MR 2 021 746.
22. O. Dubois and J. Mandler, The 3-XORSAT threshold, C R Math Acad Sci Paris 335(11) (2002), 963-966, MR 2003m:68053.
23. O. Dubois, Y. Boufkhad, and J. Mandler, Typical random 3-SAT formulae and the satisfiability threshold, 11th Annu ACM-SIAM Symp Discrete Alg, San Francisco, CA, 2000, ACM, New York, 2000, pp. 126-127.
24. O. Dubois, Y. Boufkhad, and J. Mandler, Typical random 3-SAT formulae and the satisfiability threshold, Tech Report TR03-007, Electronic Colloquium on Computational Complexity, 2003, http://www.eccc.uni-trier.de/eccc-reports/2003/TR03-007/index.html.
25. R. Durrett, Probability: Theory and examples, 2nd edition, Duxbury Press, Belmont, CA, 1996, MR 98m:60001.
26. W. Fernandez de la Vega, On random 2-SAT, manuscript, 1992.
27. W. Fernandez de la Vega, Random 2-SAT: Results and problems, Theoret Comput Sci 265(1-2) (2001), 131-146, Phase Transitions in Combinatorial Problems, Trieste, 1999, MR 2002i:68053.
28. E. Friedgut, Necessary and sufficient conditions for sharp thresholds of graph properties, and the k-SAT problem, J Amer Math Soc 12 (1999), 1017-1054.
29. J. Friedman, A proof of Alon's second eigenvalue conjecture, unpublished manuscript, http://www.math.ubc.ca/jf/pubs/, 2002.
30. J. Friedman, J. Kahn, and E. Szemerédi, On the second eigenvalue in random regular graphs, 21st Annu ACM Symp Theory Computing, ACM Press, Seattle, WA, 1989, pp. 587-598.
31. D. Gamarnik, Linear phase transition in random linear constraint satisfaction problems, Probab Theory Related Fields (2004), to appear. Preliminary version appeared in Proc 15th ACM-SIAM Symp Discrete Algorithms, New Orleans, LA, 2004, pp. 104-113.
32. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J ACM 42(6) (1995), 1115-1145, MR 97g:90108.
33. A. Goerdt, A threshold for unsatisfiability, J Comput System Sci 53(3) (1996), 469-486.
34. J. Håstad, Some optimal inapproximability results, STOC '97 El Paso, TX, ACM, New York, 1997, pp. 1-10 (electronic), MR 1 715 618.
35. S. Janson, T. Luczak, and A. Ruciński, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000, MR 2001k:05180.
36. S. Janson, Y. C. Stamatiou, and M. Vamvakari, Bounding the unsatisfiability threshold of random 3-SAT, Random Structures Algorithms 17(2) (2000), 103-116.
37. V. Kalapala and C. Moore, Max Cut on sparse random graphs, unpublished manuscript, September 2002.
38. A. C. Kaporis, L. M. Kirousis, and E. Lalas, The probabilistic analysis of a greedy satisfiability algorithm. Fifth Int Symp Theory Appl Satisfiability Testing, 2002, pp. 362-376.
39. H. Karloff and U. Zwick, A 7/8-approximation algorithm for MAX 3SAT?, Proc 38th Annu IEEE Symp Found Comput Sci, Miami Beach, FL, IEEE Press, New York, 1997, pp. 406-415.
40. J. H. Kim, Poisson cloning model for random graphs, in preparation.
41. M. Lewin, D. Livnat, and U. Zwick, Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems, Proc IPCO, 2002, pp. 67-82.
42. C. McDiarmid, On the method of bounded differences, London Mathematical Society Lecture Note Series, Vol. 141, Cambridge University Press, Cambridge, 1989, pp. 148-188.
43. J. Neveu, Discrete-parameter martingales, revised edition, North-Holland, Amsterdam, 1975, translated from the French by T. P. Speed. North-Holland Mathematical Library, Vol. 10, MR 53 #6729.
44. J. H. Spencer, Ten lectures on the probabilistic method, 2nd edition, SIAM, Philadelphia, 1994.
45. L. Trevisan, G. B. Sorkin, M. Sudan, and D. P. Williamson, Gadgets, approximation, and linear programming, SIAM J Comput 29(6) (2000), 2074-2097, MR 2001j:68046.
46. Y. Verhoeven, Random 2-SAT and unsatisfiability, Inform Process Lett 72(3-4) (1999), 119-123, MR 2000k:68074.
47. Y. C. Verhoeven, The bisection of random graphs: Rigorous bounds using the analysis of algorithms, unpublished manuscript, http://ycverhoeven.free.fr/index.en.html, 2000.
48. Y. C. Verhoeven, Quelques utilisations des arbres en combinatoire, Ph.D. thesis, Université Paris-Sud, 2001.
49. D. B. Wilson, On the critical exponents of random k-SAT, Random Structures Algorithms 21(2) (2002), 182-195.
50. N. C. Wormald, Differential equations for random processes and random graphs, Ann Appl Probab 5(4) (1995), 1217-1235.