Improved Algorithms for Maximum Satisfiability and Its Special Cases

Abstract

The Maximum Satisfiability (MAXSAT) problem is an optimization version of the Satisfiability problem (SAT) in which one is given a CNF formula with n variables and needs to find the maximum number of simultaneously satisfiable clauses. Recent works achieved significant progress in proving new upper bounds on the worst-case computational complexity of MAXSAT. All these works reduce general MAXSAT to a special case of MAXSAT where each variable appears a small number of times. So, it is important to design fast algorithms for (n,k)-MAXSAT to construct an efficient exact algorithm for MAXSAT. (n,k)-MAXSAT is a special case of MAXSAT where each variable appears at most k times in the input formula. For the (n,3)-MAXSAT problem, we design a O*(1.1749^n) algorithm improving on the previous record running time of O*(1.191^n). For the (n,4)-MAXSAT problem, we construct a O*(1.3803^n) algorithm improving on the previous best running time of O*(1.4254^n). Using the results, we develop a O*(1.0911^L) algorithm for the MAXSAT where L is a length of the input formula which improves previous algorithm with O*(1.0927^L) running time.