Abstract
We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. We show that using the relaxation, a proof of the unsatisfiability of the notorious pigeonhole and mutilated chessboard problems can be computed in polynomial time. As a byproduct we find a new `sandwich" theorem that is similar to the sandwich theorem for Lovász' ϑ-function. Furthermore, the semidefinite relaxation gives a certificate of (un)satisfiability for 2SAT problems in polynomial time. By adding an objective function to the dual formulation, a specific class of polynomially solvable 3SAT instances can be identified. We conclude with discussing how the relaxation can be used to solve more general SAT problems and with some empirical observations.
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References
Alizadeh, F.: Combinatorial optimization with interior point methods and semi-definite matrices, Ph.D. Thesis, University of Minnesota, Minneapolis, 1991.
Aspvall, B., Plass, M. F., and Tarjan, R. E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas, Inform. Process. Lett. 8(3) (1979), 121–123.
Benson, S. J., Ye, Y., and Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization, Technical Report, Computational Optimization Lab., Department of Management Science, University of Iowa, Iowa City, 1997.
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Math. 4 (1973), 305–337.
Cook, S. A.: The complexity of theorem proving procedures, in Proceedings of the 3rd annual ACM symposium on the Theory of Computing, 1971, pp. 151–158.
Cook, W., Coullard, C. R., and Turan, G.: On the complexity of cutting plane proofs, Discrete Appl. Math. 18 (1987), 25–38.
Davis, M., Logemann, M., and Loveland, D.: A machine program for theorem proving, Comm. ACM 5 (1962), 394–397.
de Klerk, E.: Interior point methods for semidefinite programming, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1997.
de Klerk, E., Pasechnik, D. V., and Warners, J. P.: Approximate graph colouring algorithms based on the #-function, Technical Report, 1999. In preparation.
de Klerk, E. and Warners, J. P.: Semidefinite programming techniques for MAX-2-SAT and MAX-3-SAT: Computational perspectives, Technical Report 98–34, Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands, 1998.
Du, D., Gu, J., and Pardalos, P. M. (eds.): Satisfiability Problem: Theory and Applications, DIMACS Series in Discrete Math. and Comput. Sci. 35, Amer. Math. Soc., 1997.
Feige, U. and Goemans, M.: Approximating the value of two prover proof systems with applications to MAX 2SAT and MAX DICUT, in Proc. Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182–189.
Goemans, M. X. and Williamson, D. P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42(6) (1995), 1115–1145.
Gu, J., Purdom, P. W., Franco, J., and Wah, B. W.: Algorithms for the satisfiability (SAT) problem: A survey, in Du et al. [11], pp. 9–151.
Haken, A.: The intractability of resolution, Theoret. Comput. Sci. 39 (1985), 297–308.
Helmberg, C. and Rendl, F.: A spectral bundle method for SDP, Technical Report ZIB, Preprint SC–97–37, Konrad-Zuse-Zentrum, Berlin, 1997.
Hooker, J. N.: Resolution vs. cutting plane solution of inference problems: some computational experience, Oper. Res. Lett. 7(1) (1988), 1–7.
Horn, R. A. and Johnson, C. R.: Matrix Analysis, Cambridge University Press, 1985.
Impagliazzo, R., Pitassi, T., and Urquhart, A.: Upper and lower bounds for treelike cutting plane proofs, in Proceedings of the 9th Annual IEEE Symposium on Logic in Computer Science, 1994, pp. 220–228.
Karloff, H. and Zwick, U.: A 7/8-approximation algorithm for MAX3SAT? in Proceedings of the 38th Symposium on the Foundations of Computer Science, 1997, pp. 406–415.
Kullmann, O.: Investigations on autark assignments, Technical Report, Johann Wolfgang Goethe-Universität, Fachbereich Mathematik, 60054 Frankfurt, Germany, 1998. Submitted.
Lovász, L.: On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 1–7.
Lovász, L. and Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization, SIAM J. Optim. 1(2) (1991), 166–190.
Van Maaren, H.: Elliptic approximations of propositional formulae, Technical Report 96–65, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1996. To appear in Discrete Appl. Math.
Van Maaren, H.: On the use of second order derivatives for the satisfiability problem, in Du et al. [11], pp. 677–687.
Van Maaren, H. and Warners, J. P.: Solving satisfiability problems using elliptic approximations-a note on volumes and weights, Technical Report 98–32, Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands, 1998.
Monien, B. and Speckenmeyer, E.: Solving satisfiability in less than 2n steps, Discrete Appl. Math. 10 (1985), 287–295.
Schaefer, T. J.: The complexity of satisfiability problems, in Proceedings of the Tenth Symposium on the Theory of Computing, 1978, pp. 216–226.
Sherali, H. D. and Adams, W. P.: A hierarchy of relaxations between the continuous and convex hull representations for 0–1 programming problems, SIAM J. Discrete Math. 3(3) (1990), 411–430.
Strang, G.: Linear Algebra and Its Applications, 3rd edn, Harcourt Brace Jovanovich, 1988.
Sturm, J. F.: Using SeDuMi 1:02, a MATLAB toolbox for optimization over symmetric cones, Technical Report, Communications Research Laboratory, McMaster University, Hamilton, Canada, 1998.
Trevisan, L., Sorkin, G., Sudan, M., and Williamson, D.: Gadgets, approximation and linear programming, in Proceedings of the 37th Symposium on the Foundations of Computer Science, 1996, pp. 617–626.
Urquhart, A.: Open Problem Posed at SAT'98, Paderborn, Germany, 1998.
Vandenberghe, L. and Boyd, S.: Semidefinite programming, SIAM Rev. 38 (1996), 49–95.
Warners, J. P. and Van Maaren, H.: Recognition of tractable satisfiability problems through balanced polynomial representations, Discrete Appl. Math., 1999. To appear.
Warners, J. P. and Van Maaren, H.: Solving satisfiability problems using elliptic approximations-effective branching rules, Technical Report 98–18, Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands, 1998. Accepted for publication in Discrete Appl. Math.
Warners, J. P. and Van Maaren, H.: A two-phase algorithm for solving a class of hard satisfiability problems, Oper. Res. Lett. 23(3–5) (1999), 81–88.
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de Klerk, E., van Maaren, H. & Warners, J.P. Relaxations of the Satisfiability Problem Using Semidefinite Programming. Journal of Automated Reasoning 24, 37–65 (2000). https://doi.org/10.1023/A:1006362203438
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DOI: https://doi.org/10.1023/A:1006362203438