## MAX-SATedited by Thomas Stützle and Luis Paquete, INTELLEKTIK ## Contents## About MAX-SATGivenn binary variables x(j), j in N, m
clauses C_i, i in
M, and weights w(i), where N={1,2,... ,n} and M={1,2,
...,m}, the MAX-SAT problem asks to determine a binary tuple
(that is, a 0-1 assignment to each of the binary variables) that
maximizes the sum of the weights of the satisfied clauses. Each clause
is a disjunction of literals l*(j), where a literal is either a
variable x(j) (that is, the variable occurs in positive form) or its
negation neg x(j) (that is, the variable occurs in negative
form). Without loss of generality we assume that at most one of x(j)
and neg x(j) is included in each clause. A clause is satisfied if at
least one of the positive variables contained in the clause is
assigned the value 1 (true) or a negated variable is assigned the
value 0 (false). Given a set of clauses, MAX-SAT is the problem to find a variable assignment that maximizes the weight of satisfied clauses. Note that alternatively, the objective function could have been defined to minimize the weight of the unsatisfied clauses, as it is often done in the literature. If all the weights are equal to one, we call the resulting problem unweighted MAX-SAT, otherwise we speak of weighted MAX-SAT or simply MAX-SAT. The unweighted MAX-SAT is important, because a special case is the SAT problem. The SAT problem is a decision problem that asks whether a binary tuple can be found that satisfies all clauses (corresponding to a solution of weight n). The SAT problem is a central problem in theoretical computer science, artificial intelligence, mathematical logic, and many applications.
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