Backbone Guided Local Search for the Weighted Maximum Satisfiability Problem

The Satisfiability problem (SAT) is a famous NP-Complete problem, which consists of an assignment of Boolean variables (true or false) and some clauses formed of these variables. A clause is a disjunction of some Boolean literals and can be true if and only if any of them is true. A SAT instance is satisfied if and only if all the clauses are simultaneously true. As a generalization of SAT, the Maximum Satisfiability problem (MAX-SAT) aims to maximize the number of satisfied clauses. When every clause is associated with some weight, the MAX-SAT turns to the weighted Maximum Satisfiability problem (weighted MAX-SAT) with numerous applications arising in artificial intelligence, such as scheduling, data mining, pattern recognition, and automatic reasoning. According to the computational complexity theory, there’s no polynomial time algorithm for solving weighted MAX-SAT unless P NP = . Hence, many heuristic algorithms capable of finding near optimal solutions in reasonable time have been proposed for weighted MAX-SAT, including GSAT, GLS, Taboo Scatter Search, ACO, and GRASP/GRASP with path-relinking. As an efficient tool for heuristic design, the backbone has attracted great attention from the society of artificial intelligence in recent years. The backbone is defined as the common parts of all optimal solutions for an instance. Since it’s usually intractable to obtain the exact backbone, many approximate backbone guided heuristics have been developed for NPComplete problems, including 3-SAT, TSP, QAP, et al. For SAT or MAX-SAT, some character and algorithms of backbone are in research. And a BGWalksat (backbone guided local search algorithm with Walksat operator) has been developed for MAX-SAT and achieves better performance than existing heuristics. However, as to our knowledge, there is no theoretical result or backbone guided heuristics reported in the literature for weighted MAX-SAT. In this chapter we investigated the computational complexity of the backbone for weighted MAX-SAT and developed a backbone guided local search (BGLS) algorithm. Firstly, we proved that there’s no polynomial time algorithm to retrieve the full backbone under the assumption that P NP ≠ . In the proof, we mapped any weighted MAX-SAT instance to a biased weighted MAX-SAT instance with a unique optimal solution by slightly perturbing, which is also optimal to the original instance. Based on this proof, we indicated that it’s also intractable to retrieve a fixed fraction of the backbone, by reducing any weighted MAX-SAT instance to a series of weighted MAX-SAT instances with smaller scale. Secondly, we O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg