A Backbone-Based Co-evolutionary Heuristic for Partial MAX-SAT Mohamed El Bachir Menaï 1 and Mohamed Batouche 2 1 Laboratoire d’Intelligence Artiﬁcielle, Université de Paris8, 2 rue de la liberté, 93526 Saint-Denis, France menai@ai.univ-paris8.fr 2 Laboratoire LIRE, Département d’Informatique, Université Mentouri, 25000 Constantine, Algérie batouche@wissal.dz Abstract. The concept of backbone variables in the satisﬁability prob- lem has been recently introduced as a problem structure property and shown to inﬂuence its complexity. This suggests that the performance of stochastic local search algorithms for satisﬁability problems can be improved by using backbone information. The Partial MAX-SAT Prob- lem (PMSAT) is a variant of MAX-SAT which consists of two CNF formulas deﬁned over the same variable set. Its solution must satisfy all clauses of the ﬁrst formula and as many clauses in the second formula as possible. This study is concerned with the PMSAT solution in setting a co-evolutionary stochastic local search algorithm guided by an estimated backbone variables of the problem. The eﬀectiveness of our algorithm is examined by computational experiments. Reported results for a number of PMSAT instances suggest that this approach can outperform state- of-the-art PMSAT techniques. 1 Introduction Many problems in artiﬁcial intelligence (AI) and operations research (OR) are optimization problems, where the objective is to ﬁnd a best assignment to a set of variables such that a set of constraints are satisﬁed. Real world problems found in application areas including scheduling [4] and pattern recognition [12] contain hard and soft constraints. Hard constraints must be satisﬁed by any solution, while soft constraints specify a function to be optimized. Various approaches have been proposed to represent over-constrained problems. Freuder and Wallace [12] presented the concept of partial constraint satisfaction, where the objective is to maximize the total number of satisﬁed constraints. Borning et al. [7] introduced the notion of constraint hierarchies, where the distinction between hard and soft constraints is extended to a multiple level constraint hierarchy. Boolean satisﬁability (SAT) is among the most interesting AI formalisms for reasoning, planning and learning [23]. The SAT problem asks to decide whether a given propositional formula, in conjunctive normal form (CNF), has a model. The maximum satisﬁability (MAX-SAT) problem is the optimization version of SAT E. Talbi et al. (Eds.): EA 2005, LNCS 3871, pp. 155–166, 2006. c  Springer-Verlag Berlin Heidelberg 2006