Efficient 2 and 3-Flip Neighborhood Search Algorithms for the MAX SAT

For problems SAT and MAX SAT, local search algorithms are widely acknowledged as one of the most e ective approaches. Most of the local search algorithms are based on the 1ip neighborhood, which is the set of solutions obtainable by ipping the truth assignment of one variable. In this paper, we consider rip neighborhoods for r 2, and propose, for r = 2; 3, new implementations that reduce the number of candidates in the neighborhood without sacri cing the solution quality. For 2ip (resp., 3ip) neighborhood, we show that its expected size is O(n +m) (resp., O(m+ t 2 n)), which is usually much smaller than the original size O(n 2 ) (resp., O(n 3 )), where n is the number of variables, m is the number of clauses and t is the maximum number of appearances of one variable. Computational results tell that these estimates by the expectation well represent the real performance. These neighborhoods are then used under the framework of tabu search etc., and compared with other existing algorithms based on 1ip neighborhood. The results exhibit good prospects of the proposed algorithms.