On semidefinite programming relaxations for the satisfiability problem

Anjos, Miguel F. (2004) On semidefinite programming relaxations for the satisfiability problem. Mathematical Methods of Operations Research, 60 (3), 349-367. (doi:10.1007/s001860400377).

Record type: Article

Abstract

This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in conjunctive normal form. We propose a new SDP relaxation for 3-SAT and prove some of its theoretical properties. We also show that, together with two SDP relaxations previously proposed in the literature, the new relaxation completes a trio of linearly sized relaxations with increasing rank-based guarantees for proving satisfiability. A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm.

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Published date: 2004

Additional Information: 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 must be positive semidefinite. This paper is concerned with the construction and theoretical properties of semidefinite relaxations for SAT which are computationally efficient. We focus on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT. We propose a new semidefinite relaxation for 3-SAT, prove some of its theoretical properties, and show how it completes a trio of relaxations with increasing rank-based guarantees of optimality. We also present computational results showing that a basic enumerative algorithm using this new relaxation and an appropriate SDP solver is able to prove either satisfiability or unsatisfiability of 3-SAT instances with several hundred clauses.

Keywords: satisfiability problem, semidefinite programming, combinatorial optimization, global optimization

Organisations: Operational Research