MAX-SAT

In computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean formula in conjunctive normal form, that can be made true by an assignment of truth values to the variables of the formula. It is a generalization of the Boolean satisfiability problem, which asks whether there exists a truth assignment that makes all clauses true.

The conjunctive normal form formula

is not satisfiable: no matter which truth values are assigned to its two variables, at least one of its four clauses will be false. However, it is possible to assign truth values in such a way as to make three out of four clauses true; indeed, every truth assignment will do this. Therefore, if this formula is given as an instance of the MAX-SAT problem, the solution to the problem is the number three.

The MAX-SAT problem is NP-hard, since its solution easily leads to the solution of the boolean satisfiability problem, which is NP-complete.

It is also difficult to find an approximate solution of the problem, that satisfies a number of clauses within a guaranteed approximation ratio of the optimal solution. More precisely, the problem is APX-complete, and thus does not admit a polynomial-time approximation scheme unless P = NP.

Many exact solvers for MAX-SAT have been developed during recent years, and many of them were presented in the well-known conference on the boolean satisfiability problem and related problems, the SAT Conference. In 2006 the SAT Conference hosted the first MAX-SAT evaluation comparing performance of practical solvers for MAX-SAT, as it has done in the past for the pseudo-boolean satisfiability problem and the quantified boolean formula problem. Because of its NP-hardness, large-size MAX-SAT instances cannot be solved exactly, and one must resort to approximation algorithms and heuristics

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