2. The maximum satisfiability problem

Given a set of n Boolean variables and a conjunctive normal form (CNF) of a set of m disjunctive clauses of literals, where each literal is a variable or its negation which takes one of the two values True or False, the task is to determine whether there exists an assignment of truth values of the variables that satisfy the maximum number k of clauses. Multilevel approaches are special techniques which aim at producing smaller and smaller problems that are easier to solve than the original one. These techniques were applied to different combinatorial optimization problems. Examples include graph-partitioning problem [3–7], the traveling salesman problem [8, 9], graph coloring and graph drawing [10, 11], feature selection problem in biomedical data [12], and maximum satisfiability problem [13–16]. A recent survey over multilevel techniques can be found in [1, 17, 18].
