Walksat in the 2004 SAT Competition

The first convincing demonstration that local search could be used to solve challenging satisfiability problems was provided by the GSAT algorithm [9]. GSAT performs gradient descent search in the space of complete truth assignments, where adjacent assignments differ on a single variable and the objective function is the number of clauses not satisfied by the assignment. Like all local search routines GSAT can become trapped in a local minima. One technique for reducing this problem is to randomly alternate between greedy minimizing moves and “noisy” moves that are randomly selected from the variables that appear in unsatisfied clauses [7]. The Walksat algorithm [8] is based on the insight that such noisy moves could be made the basis for local search. Rather than trying to globally determine the best move, Walksat first randomly chooses an unsatisfied clause, and then selects a variable to flip within the clause. Because Walksat may overlook the best global move it is said to perform hill-climbing rather than gradient descent. The fact that a clause is unsatisfied means that at least one of the variables in the clause must be flipped in order to reach a global solution. If variables are chosen randomly from the clause and the clause length is bounded by a constant k it is easy to see that each flip as a 1/k or better chance of being correct. When k = 2 a pure random walk strategy will solve a satisfiable formula over n variables with high probability in O(n) time [5]. For larger values of k the only worst-case guarantees for pure random walk are exponential. In practice, therefore, the variable to be flipped is chosen from the (randomly selected) unsatisfied clause by some greedy heuristic. A number of such heuristics have been studied. The original Walksat heuristic, denoted “SKC” after the initials of the authors [8], employs the notion of the breakcount of a variable, which is the number of clauses that are currently satisfied that would become unsatisfied if the variable were to be flipped. Similarly, the makecount of a variable is the number of clauses current unsatisfied that would become satisfied. The SKC variable selection heuristic is as follows: (1) If there are variables with breakcount=0, choose one such variable at random. (2) Otherwise, with some fixed probability p select a variable randomly from the clause. (3) Otherwise, pick a variable with minimum breakcount; if there are several such, pick one at random. The first step, checking for any breakcounts of 0, is crucial for good performance of the SKC heuristic. If this step is eliminated and variables are chosen to minimize (breakcount makecount), then the algorithm is similar to GSAT, modulo the fact that variable selection is not done globally, but only after clause selection. The breakcounts and makecounts of all variables can be maintained incrementally, an implementation decision that is crucial for practical efficiency of Walksat. At the start of the algorithm the values are initialized. When a variable x is flipped from true to false, the breakcounts can be updated by (i) visiting each clause that contains x positively; and if the clause now contains single true literal l, incrementing the breakcount of the variable of l; and (ii) visiting each clause containing x negatively; and if the clause now contains exactly one other true literal l, decrementing the the breakcount of the variable of l. Similar operations are performed for the case of flipping from false to true or cal-