Satisfying versus Falsifying in Local Search for Satisfiability - (Poster Presentation)

During local search, clauses may frequently be satisfied or falsified. Modern SLS algorithms often exploit the falsifying history of clauses to select a variable to flip, together with variable properties such as score and age. The score of a variable x refers to the decrease in the number of unsatisfied clauses if x is flipped. The age of x refers to the number of steps done since the last time when x was flipped. Novelty [5] and Novelty based SLS algorithms consider the youngest variable in a randomly chosen unsatisfied clause c, which is necessarily the last falsifying variable of c whose flipping made c from satisfied to unsatisfied. If the best variable according to scores in c is not the last falsifying variable of c, it is flipped, otherwise the second best variable is flipped with probability p, and the best variable is flipped with probability 1-p. TNM [4] extends Novelty by also considering the second last falsification of c, the third last falsification of c, and so on... If the best variable in c most recently and consecutively falsified c several times, TNM considerably increases the probability to flip the second best variable of c. Another way to exploit the falsifying history of clauses is to define the weight of a clause to be the number of local minima in which the clause is unsatisfied, so that the objective function is to reduce the total weight of unsatisfied clauses. In this paper, we propose a new heuristic by considering the satisfying history of clauses instead of their falsifying history, and by modifying Novelty as follows: If the best variable in c is not the most recent satisfying variable of c, flip it. Otherwise, flip the second best variable with probability p, and flip the best variable with probability 1-p. Here, the most recent satisfying variable in c is the variable whose flipping most recently made c from unsatisfied to satisfied. The intuition of the new heuristic is to avoid repeatedly satisfying c using the same variable. Note that in a clause c, the most recent falsifying variable and the most recent satisfying variable can be the same variable. In this case, the variable flipped to make c from unsatisfied to satisfied was re-flipped later to make c from satisfied to unsatisfied (there can be other flips between the two flips), and vice versa. In our experiments using instances from the 2011 SAT competition, this is the case in the randomly selected unsatisfied clause in more than 95% steps for random 3-SAT. The percentage is less than 90% for random 5-SAT and 7-SAT, and for crafted instances. So the new heuristic is expected to behave similarly as Novelty on random 3-SAT, but differently for other SAT problems. We propose a new SLS algorithm called SatTime that implements the new heuristic. Given a SAT instance φ to solve, SatTime first generates a random assignment and while the assignment does not satisfy φ, it repeatedly modifies the assignment as follows: