Low-rank semidefinite programming for the MAX2SAT problem

This paper proposes a new algorithm for solving MAX2SAT problems based on combining search methods with semidefinite programming approaches. Semidefinite programming techniques are well-known as a theoretical tool for approximating maximum satisfiability problems, but their application has traditionally been very limited by their speed and randomized nature. Our approach overcomes this difficult by using a recent approach to low-rank semidefinite programming, specialized to work in an incremental fashion suitable for use in an exact search algorithm. The method can be used both within complete or incomplete solver, and we demonstrate on a variety of problems from recent competitions. Our experiments show that the approach is faster (sometimes by orders of magnitude) than existing state-of-the-art complete and incomplete solvers, representing a substantial advance in search methods specialized for MAX2SAT problems.Comment: Accepted at AAAI'19. The code can be found at https://github.com/locuslab/mixsa

Sums of squares based approximation algorithms for MAX-SAT

AbstractWe investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem. To be specific, we examine the potential of this theory for providing tests for unsatisfiability and providing MAX-SAT upper bounds. We compare the SOS approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42(6) (1995) 1115–1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182–189] and the MAX-3-SAT case of Karloff and Zwick [A 7/8-approximation algorithm for MAX 3SAT? in: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, USA, IEEE Press, New York, 1997], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is an SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS Semidefinite Program (SDP) which we experimentally compare with the appropriate existing rounding techniques. Further we investigate the implications to the decision variant SAT and compare experimental results with those yielded from the higher lifting approach of Anjos [On semidefinite programming relaxations for the satisfiability problem, Math. Methods Oper. Res. 60(3) (2004) 349–367; An improved semidefinite programming relaxation for the satisfiability problem, Math. Programming 102(3) (2005) 589–608; Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems, J. Satisfiability Boolean Modeling Comput. 1 (2005) 1–47].We give some impression of the fraction of the so-called unit constraints in the various SDP relaxations. From a mathematical viewpoint these constraints should be easily dealt within an algorithmic setting, but seem hard to be avoided as extra constraints in an SDP setting. Finally, we briefly indicate whether this work could have implications in finding counterexamples to uncovered cases in Hilbert's Positivstellensatz