Operations Research Letters 36 (2008) 434–438 www.elsevier.com/locate/orl Solving weighted MAX-SAT via global equilibrium search Oleg V. Shylo a , Oleg A. Prokopyev b,∗ , Vladimir P. Shylo c a Department of ISE, University of Florida, Gainesville, FL 32611, USA b Department of IE, University of Pittsburgh, Pittsburgh, PA 15232, USA c Institute of Cybernetics, NAS of Ukraine, Kiev, Ukraine Received 15 January 2007; accepted 27 November 2007 Available online 31 January 2008 Abstract In this note we investigate the performance of global equilibrium search based heuristics on the weighted MAX-SAT problem. Three variants of the approach are implemented and compared with other existing algorithms on publicly available benchmark instances. The reported computational results indicate high efficiency of the method considered. c  2008 Elsevier B.V. All rights reserved. Keywords: Weighted maximum satisfiability; Global equilibrium search; Heuristics 1. Introduction Let X ={x 1 , x 2 ,..., x n } be a set of Boolean variables, i.e., x i can take the value 0 (false) or the value 1 (true). A propositional formula Φ is said to be in conjunctive normal form (CNF) if it is a conjunction of clauses C = {C 1 , C 2 ,..., C m }, where each clause C i is a disjunction of literals: Φ = m  i =1  |C i |  j =1 ℓ ij  , (1) where |C i | is the number of literals in a clause C i , and ℓ ij is a literal, i.e., a Boolean variable x k , or its negation ¯ x k ,1 ≤ k ≤ n. A clause is satisfied if at least one of its literals is true. In the MAXIMUM SATISFIABILITY (MAX-SAT) problem we need to find an assignment of values to the variables that satisfies as many clauses as possible. A natural generalization of the above problem is to define a positive weight w i for each clause C i and search for an assignment, which maximizes the total weight of the satisfied clauses. The MAX-SAT as well as its weighted version remain NP - hard even if each clause has at most two literals (MAX-2SAT problem) [5,11]. ∗ Corresponding address: 1037 Benedum Hall, University of Pittsburgh, Pittsburgh, PA 15261, USA. E-mail address: prokopyev@engr.pitt.edu (O.A. Prokopyev). Among various heuristic approaches for solving the MAX- SAT problem we should mention algorithms based on reactive tabu search [1], simulated annealing [22], GRASP [4,18, 19], iterated local search [25] and guided local search [12]. Detailed surveys on MAX-SAT, related applications and solution approaches can be found in [2,3,6,9,10]. Some of the algorithmic codes and benchmark instances are publicly available and can be found at [16,17,23,26]. In this note we consider a heuristic approach for solving weighted MAX-SAT based on the global equilibrium search (GES) framework suggested in [20,21]. We investigate the performance of three variants of the proposed method and compare them with other existing algorithms on publicly available benchmark instances. The computational results revealed strong and robust performance of the approach asserting that the techniques similar to the simulated annealing provide a competitive solving tool for weighted MAX-SAT. Finally, we believe that GES based methods are promising for solving a variety of discrete optimization problems, which is also confirmed by our work in application of GES for the unconstrained quadratic 0–1 programming problem [14]. 2. GES for the weighted MAX-SAT The global equilibrium search algorithm (GES) uses concepts which are similar to those of the simulated annealing (SA) method. A detailed description of GES is given in [20,21]. 0167-6377/$ - see front matter c  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2007.11.007