Operations Research Proceedings 2003 pp 308-315 | Cite as

# Proofs of Unsatisfiability Via Semidefinite Programming

## Abstract

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable *X* is maximized (or minimized) subject to linear constraints on the elements of *X* and the additional constraint that *X* be positive semidefinite. We focus on the application of SDP to obtain proofs of unsatisfiability. Using a new SDP relaxation for SAT, we obtain proofs of unsatisfiability for some hard instances with up to 260 variables and over 400 clauses. In particular, we can prove the unsatisfiability of the smallest unsatisfiable instance that remained unsolved during the SAT Competition 2003. This shows that the SDP relaxation is competitive with the top solvers in the competition, and that this technique has the potential to complement existing techniques for SAT.

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### References

- 1.M.F. Anjos.
*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.Google Scholar - 2.M.F. Anjos. An improved semidefinite programming relaxation for the satisfiability problem.
*Math. Program*. (Ser. A), to appear, 2004.Google Scholar - 3.M.F. Anjos and H. Wolkowicz. Semidefinite programming for discrete optimization and matrix completion problems.
*Discrete Appl. Math.*, 123(1–2):513–577, 2002.Google Scholar - 4.M.F. Anjos and H. Wolkowicz. Strengthened semidefinite relaxations via a second lifting for the max-cut problem.
*Discrete Appl. Math.*, 119(1–2):79–106, 2002.CrossRefGoogle Scholar - 5.E. Balas, S. Ceria, and G. Cornuéjols. A lift-and-project cutting plane algorithm for mixed 0–1 programs.
*Math. Program.*, 58(3, Ser. A):295–324, 1993.CrossRefGoogle Scholar - 6.D. Bienstock and M. Zuckerberg. Subset algebra lift operators for 0–1 integer programming. Technical Report CORC 2002-01, Columbia University, July 2002.Google Scholar
- 7.E. de Klerk and H. Van Maaren. On semidefinite programming relaxations of (2+
*p*)-SAT.*Ann. Math. Artif. Intell.*, 37(3):285–305, 2003.CrossRefGoogle Scholar - 8.E. de Klerk, H. Van Maaren, and J.P. Warners. Relaxations of the satisfiability problem using semidefinite programming.
*J. Automat. Reason.*, 24(1–2):37–65, 2000.CrossRefGoogle Scholar - 9.J. Gu, P.W. Purdom, J. Franco, and B.W. Wah. Algorithms for the satisfiability (SAT) problem: a survey. In:
*Satisfiability problem: theory and applications (Piscataway, NJ, 1996)*, pages 19–151. Amer. Math. Soc, Providence, RI, 1997.Google Scholar - 10.J.B. Lasserre. Global optimization with polynomials and the problem of moments.
*SIAM J. Optim.*, 11(3):796–817 (electronic), 2000/01.CrossRefGoogle Scholar - 11.J.B. Lasserre. An explicit equivalent positive semidefinite program for nonlinear 0–1 programs.
*SIAM J. Optim.*, 12(3):756–769 (electronic), 2002.CrossRefGoogle Scholar - 12.M. Laurent and F. Rendl. Semidefinite programming and integer programming. In: G. Nemhauser K. Aardal and R. Weismantel, editors,
*Handbook on discrete optimization*, to appear.Google Scholar - 13.L. Lovàsz and A. Schrijver. Cones of matrices and set-functions and 0–1 optimization.
*SIAM J. Optim.*, 1(2):166–190, 1991.CrossRefGoogle Scholar - 14.P.A. Parrilo. Semidefinite programming relaxations for semialgebraic problems.
*Math. Program.*, 96(2, Ser. B):293–320, 2003.CrossRefGoogle Scholar - 15.H.D. Sherali and W.P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems.
*SIAM J. Discrete Math.*, 3(3):411–430, 1990.CrossRefGoogle Scholar - 16.K.C. Toh, M.J. Todd, and R.H. Tütüncü. SDPT3—a MATLAB software package for semidefinite programming, version 1.3.
*Optim. Methods Softw.*, 11/12(1–4):545–581, 1999.CrossRefGoogle Scholar - 17.H. van Maaren. Elliptic approximations of propositional formulae.
*Discrete Appl. Math.*, 96/97:223–244, 1999.CrossRefGoogle Scholar - 18.H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors.
*Handbook of semidefinite programming*. Kluwer Academic Publishers, Boston, MA, 2000.Google Scholar