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| In [[computational complexity theory]], the '''maximum satisfiability problem''' ('''MAX-SAT''') is the problem of determining the maximum number of clauses, of a given [[Propositional formula|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.
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| ==Example==
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| The conjunctive normal form formula
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| :<math> (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)</math>
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| is not satisfiable: no matter which truth values are assigned to its two variables, at least one of its four clauses will be false.
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| 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.
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| Therefore, if this formula is given as an instance of the MAX-SAT problem. the solution to the problem is the number three.
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| ==Hardness==
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| The MAX-SAT problem is [[NP-hard]], since its solution easily leads to the solution of the [[boolean satisfiability problem]], which is [[NP-complete]].
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| It is also difficult to find an [[approximation algorithm|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.<ref>Mark Krentel. The Complexity of Optimization Problems. Proc. of STOC '86. 1986.</ref><ref>Christos Papadimitriou. Computational Complexity. Addison-Wesley, 1994.</ref><ref>Cohen, Cooper, Jeavons. A complete characterization of complexity for boolean constraint optimization problems. CP 2004.</ref>
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| ==Solvers==
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| 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 [[0-1 integer programming|pseudo-boolean satisfiability]] problem and the [[quantified boolean formula]] problem.
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| Because of its NP-hardness, large-size MAX-SAT instances cannot be solved exactly, and one must resort to [[approximation algorithm]]s
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| and [[Metaheuristic|heuristics]] <ref>
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| R. Battiti and M. Protasi.
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| Approximate Algorithms and Heuristics for MAX-SAT
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| Handbook of Combinatorial Optimization, Vol 1, 1998, 77-148, Kluwer Academic Publishers.</ref>
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| There are several solvers submitted to the last Max-SAT Evaluations:
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| * [[Branch and Bound]] based: Clone, MaxSatz (based on [[Satz (SAT solver)|Satz]]), IncMaxSatz, IUT_MaxSatz, WBO, GIDSHSat.
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| * Satisfiability based: SAT4J, QMaxSat.
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| * Unsatisfiability based: msuncore, WPM1, PM2.
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| ==Special cases==
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| 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 [[Propositional formula|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.
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| ==Related problems==
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| There are several extensions to MAX-SAT:
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| * 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.
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| * 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.
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| * 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.<ref>Josep Argelich and Felip Manyà. [http://www.springerlink.com/content/870v1535q0h51717/ Exact Max-SAT solvers for over-constrained problems]. In Journal of Heuristics 12(4) pp. 375-392. Springer, 2006.</ref>
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| * The minimum satisfiability problem.
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| * 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. <ref> {{cite journal|last1=Jaulin|first1=L.|last2=Walter|first2=E.| title=Guaranteed robust nonlinear minimax estimation| journal=IEEE Transaction on Automatic Control|year=2002|volume=47| url=http://www.ensta-bretagne.fr/jaulin/paper_qminimax.pdf}} </ref>
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| == See also ==
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| * [[Boolean satisfiability problem|Boolean Satisfiability Problem]]
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| * [[Constraint satisfaction]]
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| * [[Satisfiability Modulo Theories]]
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| == External links ==
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| * http://www.satisfiability.org/
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| * http://www.iiia.csic.es/~maxsat06
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| * http://www.maxsat.udl.cat
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| * [http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/max-sat-benchmarks.htm Weighted Max-2-SAT Benchmarks with Hidden Optimum Solutions]
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| * [http://www.cs.tau.ac.il/~azar/Methods-Class6.pdf Lecture Notes on MAX-SAT Approximation]
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| == References ==
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| <references />
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| [[Category:Logic in computer science]]
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| [[Category:Combinatorial optimization]]
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| [[Category:Satisfiability problems]]
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Jayson Berryhill is how I'm known as and my wife doesn't like it at all. Ohio is exactly where my house is but my husband desires us to transfer. To perform lacross is 1 of the things she loves most. My working day job is a travel agent.
Here is my blog post; real psychic