Probabilistic performance of a heuristic for the satisfiability problem

Abstract

An algorithm for the satisfiability problem (SAT) is presented and its probabilistic behavior is analyzed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamically assigns values to literals appearing in a given instance until a satisfying assignment is found or the algorithm “gives up” without determining whether or not a solution exists. It is shown that if n clauses are constructed independently from r Boolean variables, where the probability that a variable appears in a clause as a positive literal is p and as a negative literal is p, then almost all randomly generated instances of SAT are solved in polynomial time if p< 0.4 ln(n) r or p> ln(n) r or p= c ln(n) r , 0.4< c<<1 and lim n,r→∞ n 1−c r 1−ε <∞ for any ε>;0. It is also shown that if p= c ln(n) r , 0.4< c<1 and lim n,r→∞ n 1−c r=∞ then almost all randomly generated instances of SAT have no solution. Thus the combined algorithm is very effective in the probabilistic sense on instances of SAT that have solutions. The combined algorithm is effective in some limited sense in verifying unsatisfiability.