Maximum satisfiability problem

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.

Example

The conjunctive normal form formula

$(x_{0}\lor x_{1})\land (x_{0}\lor \lnot x_{1})\land (\lnot x_{0}\lor x_{1})\land (\lnot x_{0}\lor \lnot x_{1})$

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.

Hardness

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.^[1][2][3]

Solvers

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 ^[4]

There are several solvers submitted to the last Max-SAT Evaluations:

• Branch and Bound based: Clone, MaxSatz (based on Satz), IncMaxSatz, IUT_MaxSatz, WBO, GIDSHSat.
• Satisfiability based: SAT4J, QMaxSat.
• Unsatisfiability based: msuncore, WPM1, PM2.

Special cases

MAX-SAT is one of the optimization extensions of the boolean satisfiability problem, which is the problem of determining whether the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE. If the clauses are restricted to have at most 2 literals, as in 2-satisfiability, we get the MAX-2SAT problem. If they are restricted to at most 3 literals per clause, as in 3-satisfiability, we get the MAX-3SAT problem.

Related problems

There are several extensions to MAX-SAT:

• The weighted maximum satisfiability problem (Weighted MAX-SAT) asks for the maximum weight which can be satisfied by any assignment, given a set of weighted clauses.
• The partial maximum satisfiability problem (PMAX-SAT) asks for the maximum number of clauses which can be satisfied by any assignment of a given subset of clauses. The rest of the clauses must be satisfied.
• The soft satisfiability problem (soft-SAT), given a set of SAT problems, asks for the maximum number of sets which can be satisfied by any assignment.^[5]
• The minimum satisfiability problem.
• The MAX-SAT problem can be extended to the case where the variables of the constraint satisfaction problem belong the set of reals. The problem amounts to finding the smallest q such that the q-relaxed intersection of the constraints is not empty. ^[6]

See also

• Boolean Satisfiability Problem
• Constraint satisfaction
• Satisfiability modulo theories

External links

• http://www.satisfiability.org/
• http://www.iiia.csic.es/~maxsat06
• http://www.maxsat.udl.cat
• Weighted Max-2-SAT Benchmarks with Hidden Optimum Solutions
• Lecture Notes on MAX-SAT Approximation

References

1. Mark Krentel. The Complexity of Optimization Problems. Proc. of STOC '86. 1986.
2. Christos Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
3. Cohen, Cooper, Jeavons. A complete characterization of complexity for boolean constraint optimization problems. CP 2004.
4. R. Battiti and M. Protasi. Approximate Algorithms and Heuristics for MAX-SAT Handbook of Combinatorial Optimization, Vol 1, 1998, 77-148, Kluwer Academic Publishers.
5. Josep Argelich and Felip Manyà. Exact Max-SAT solvers for over-constrained problems. In Journal of Heuristics 12(4) pp. 375-392. Springer, 2006.
6. Jaulin, L.; Walter, E. (2002). "Guaranteed robust nonlinear minimax estimation" (PDF). IEEE Transaction on Automatic Control. 47.