A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems

We present a simple probabilistic algorithm for solving kSAT, and more generally, for solving constraint satisfaction problems (CSP). The algorithm follows a simple localsearch paradigm (cf. [9]): randomly guess an initial assignment and then, guided by those clauses (constraints) that are not satisfied, by successively choosing a random literal from such a clause and flipping the corresponding bit, try to find a satisfying assignment. If no satisfying assignment is found after O(n) steps, start over again. Our analysis shows that for any satisfiable k-CNF formula with n variables this process has to be repeated only t times, on the average, to find a satisfying assignment, where t is within a polynomial factor of (2(1 1=k)). This is the fastest (and also the simplest) algorithm for 3-SAT known up to date. We consider also the more general case of a CSP with n variables, each variable taking at most d values, and constraints of order l, and analyze the complexity of the corresponding (generalized) algorithm. It turns out that any CSP can be solved with complexity at most (d (1 1=l) + "). 1. Algorithms for k-SAT Several algorithms have been designed for k-SAT, and some in particular for the special case 3-SAT which beat the naive 2 bound that is obtained by trying all potential 2 many assignments for the n variables in the input formula. The following list summarizes the known results for k-SAT and adds our new one, indicated by [*]. A constant c in the list means that there is an algorithm of the given type (deterministic or probabilistic) with complexity within a polynomial factor of c. 3-SAT 4-SAT 5-SAT 6-SAT type ref. 1:849 det. [15] 1:782 1.835 1.867 1.888 det. [13] 1:618 1.839 1.928 1.966 det. [10] 1:588 1.682 1.742 1.782 prob. [13] 1:579 det. [17] 1:505 det. [8] 1:5 1.6 1.667 1.715 prob. [20] 1:497 det. [19] 1:476 det. [16] 1:447 1.496 1.569 1.637 prob. [14] 1:362 1.476 prob. [14] 1:334 1.5 1.6 1.667 prob. [*]