An improved semidefinite programming relaxation for the satisfiability problem

Mathematical Programming

Volumn 102, Issue 3, 2005, Pages 589-608

  Anjos, Miguel F ^a,b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Discrete optimization; Satisfiability; Semidefinite programming

Indexed keywords

ALGORITHMS; ARTIFICIAL INTELLIGENCE; FUNCTIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; OPTIMIZATION;

DISCRETE OPTIMIZATION; GAP RELAXATION; SATISFIABILITY; SEMIDEFINITE PROGRAMMING (SDP);

PROBLEM SOLVING;

EID: 17444383147     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10107-003-0495-2     Document Type: Article

References (43)

1. Anjos, M.F.: New Convex Relaxations for the Maximum Cut and VLSI Layout Problems. PhD thesis, University of Waterloo, 2001, Published online at http://etd.uwaterloo.ca/etd/manjos2001.pdf
2. Anjos, M.F., Wolkowicz, H.: Geometry of semidefinite max-cut relaxations via matrix ranks. J. Comb. Optim. 6 (3), 237-270 (2002)
3. Anjos, M.F., Wolkowicz, H.: Semidefinite programming for discrete optimization and matrix completion problems. Discr. App. Math. 123 (1-2), 513-577 (2002)
4. Anjos, M.F., Wolkowicz, H.: Strengthened semidefinite relaxations via a second lifting for the max-cut problem. Discr. App. Math. 119 (1-2), 79-106 (2002)
5. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Programming 58 (3, Ser. A), 295-324 (1993)
6. Bauschke, H.H. Kruk, S.G.: The method of reflection-projection for convex feasibility problems with an obtuse cone. Technical report, Oakland University, Rochester MI, U.S.A., February 2002
7. Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10 (2), 443-461 (electronic), (2000)
8. Bienstock, D., Zuckerberg, M.: Subset algebra lift operators for 0-1 integer programming. Technical Report CORC 2002-01, Columbia University, July 2002
9. Burer, S.: Semidefinite programming in the space of partial positive semidefinite matrices. Technical report, Department of Management Sciences, University of Iowa, May 2002
10. Conforti, M., Cornuéjols, G.: A class of logic problems solvable by linear programming. J. Assoc. Comput. Mach. 42 (5), 1107-1113 (1995)
11. de Klerk, E., van Maaren, H.: On semidefinite programming relaxations of (2 + p)-SAT. Ann. Math. Artificial Intelligence 37 (3), 285-305 (2003)
12. de Klerk, E., van Maaren, H., Warners, J.P.: Relaxations of the satisfiability problem using semidefinite programming. J. Automat. Reason. 24 (1-2), 37-65 (2000)
13. Ferris, M., Mesnier, M., Moré, J.: NEOS and Condor: Solving optimization problems over the Internet. ACM Trans. Math. Softw. 26 (1), 1-18 (2000)
14. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: General framework. SIAM J. Optim., 11 3 (01), 647-674 (electronic), (2000)
15. Goemans, M.X., Williamson, D.P.: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM J. Disc. Math. 7 (4), 656-666 (1994)
16. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42 (6), 1115-1145 (1995)
17. Grigoriev, D., Hirsch, E.A., Pasechnik, D.V.: Complexity of semialgebraic proofs. Moscow Math. J. 2 (4), 647-679 (2002)
18. Gu, J.: Global optimization for satisfiability (SAT) problem. IEEE Trans. on Know, and Data Eng. 6 (3), 361-381 (1994)
19. Gu, J., Purdom, P.W., Franco, J., Wan, B.W.: Algorithms for the satisfiability (SAT) problem: a survey. In: Satisfiability problem: theory and applications (Piscataway, NJ, 1996), Amer. Math. Soc., Providence, RI, 1997, pp. 19-151
20. Halperin, E., Zwick, U.: Approximation algorithms for MAX 4-SAT and rounding procedures for semidefinite programs. J. Algorithms 40 (2), 184-211 (2001)
21. Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. Technical Report ZR-01-26, Konrad-Zuse-Zentrum Berlin, October 2001. To appear in the Padberg Festschrift "The Sharpest Cut", SIAM
22. Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. Technical Report ZR-01-26, Konrad-Zuse-Zentrum Berlin, October 2001. To appear in the Padberg Festschrift "The Sharpest Cut", SIAM
23. Helmberg, C., Kiwiel, K.C.: A spectral bundle method with bounds. Math Program 93 (2 Ser A) 173-194 (2002)
24. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10 (3), 673-696 (electronic), (2000)
25. Henrion, D., Lasserre, J.B.: Gloptipoly - global optimization over polynomials with matlab and sedumi. Technical report, LAAS-CNRS, Toulouse, France, 2002
26. Karloff, H., Zwick, U.: A 7/8-approximation algorithm for MAX 3SAT? In: Proc. of 38th FOCS, 1997, pp. 406-415
27. Kočvara, M., Stingl, M.: PENNON - a generalized augmented lagrangian method for semidefinite programming. Technical Report 286, Institute of Applied Mathematics, University of Erlangen, 2001
28. Kočvara, M., Stingl, M.: PENNON - a code for convex nonlinear and semidefinite programming. Technical Report 290, Institute of Applied Mathematics, University of Erlangen, 2002
29. Lasserre, J.B.: Optimality conditions and LMI relaxations for 0-1 programs. Technical report, LAAS-CNRS, Toulouse, France, 2000
30. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (3), 796-817 (electronic), (2000/01)
31. Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. Optim. 12 (3), 756-769 (electronic), (2002)
32. Laurent, M.: Semidefinite relaxations for max-cut. In: M. Grötschel, (ed.), The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM, to appear
33. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 (1), 1-7 (1979)
34. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (2), 166-190 (1991)
35. Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results. Math. Prog. 95, 303-327 (2003)
36. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Programming 96 (2, Ser. B), 293-320 (2003)
37. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math. 3 (3), 411-430 (1990)
38. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12 (1-4), 625-653 (1999)
39. Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3 - a MATLAB software package for semidefinite programming, version 1. 3. Optim. Methods Softw. 11/12 (1-4), 545-581 (1999)
40. van Maaren, H.: Elliptic approximations of prepositional formulae. Discrete Appl. Math. 96/97, 223-244 (1999)
41. Warners, J.P., van Maaren, H.: A two-phase algorithm for solving a class of hard satisfiability problems. Oper. Res. Lett. 23 (3-5), 81-88 (1998)
42. Wolkowicz, H., Saigal, R., vandenberghe, L., editors.: Handbook of semidefinite programming. Kluwer Academic Publishers, Boston, MA, 2000
43. Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of 31 st STOC, 1999, pp. 679-687