A Hyper-Heuristic method for MAX-SAT

In this paper, we are interested in the Maximum Satisfiability Problem (MAX-SAT) which is an optimization variant of the Boolean satisfiability problem (SAT). SAT is of a central importance in various areas of computer science, including theoretical computer science, algorithmic, artificial intelligence, hardware design and verification. Formally, given a set of m clauses C = {C1;C2 . . . Cm} involving a set of n Boolean variables X = {X1;X2 . . . Xn} where a clause is a disjunction of literals and a literal is a variable or its negation, the SAT problem [1] is to decide whether an assignment of truth values to the variables of X exists or not such that all the clauses of C are simultaneously satisfied. Given a propositional formula F expressed in conjunctive normal form (CNF), the MAX-SAT problem consists in finding a variable truth assignment that maximizes the number of satisfied clauses of F . MAX-SAT is NP-Hard even when each clause has no more than two literals, while SAT with two literals per clause can be solved in polynomial time. In this work, we investigate a hyper-heuristic approach for MAX-SAT. A hyper-heuristic is a high-level method that incorporates a set of low-level heuristics to handle classes of problems rather than solving one problem. The hyper-heuristic method allows to select automatically during the search process the heuristic that should be applied for finding good quality solutions and in this way avoid search stagnation. The low-level heuristics can be either constructive or perturbative heuristics. The constructive hyper-heuristics use a set of constructive heuristics that start with an empty solution and try to complete it at each step while the perturbative hyper-heuristics start with a complete initial solution and try to find better ones by improving it. In general, a hyper-heuristic works as follow: Given an instance of a problem, the high level method uses a selection criterion or a choice function strategy to choose the adequate low-level heuristic at any given time during the search. In this work, we develop a hyper-heuristic for the MAX-SAT problem. The proposed approach performs a hybrid selection strategy that makes a balance between a choice function and randomness. These two components of the selection strategy of the proposed hyper-heuristic are controlled by using a walk probability wp as it is done in a classical stochastic local search.