Penalty Formulations and Trap-Avoidance Strategies for Solving Hard Satisfiability Problems

In this paper we study the solution of SAT problems formulated as discrete decision and discrete constrained optimization problems. Constrained formulations are better than traditional unconstrained formulations because violated constraints may provide additional forces to lead a search towards a satisfiable assignment. We summarize the theory of extended saddle points in penalty formulations for solving discrete constrained optimization problems and the associated discrete penalty method (DPM). We then examine various formulations of the objective function, choices of neighborhood in DPM, strategies for updating penalties, and heuristics for avoiding traps. Experimental evaluations on hard benchmark instances pinpoint that traps contribute significantly to the inefficiency of DPM and force a trajectory to repeatedly visit the same set of or nearby points in the original variable space. To address this issue, we propose and study two trap-avoidance strategies. The first strategy adds extra penalties on unsatisfied clauses inside a trap, leading to very large penalties for unsatisfied clauses that are trapped more often and making these clauses more likely to be satisfied in the future. The second strategy stores information on points visited before, whether inside traps or not, and avoids visiting points that are close to points visited before. It can be implemented by modifying the penalty function in such a way that, if a trajectory gets close to points visited before, an extra penalty will take effect and force the trajectory to a new region. It specializes to the first strategy because traps are special cases of points visited before. Finally, we show experimental results on evaluating benchmarks in the DIMACS and SATLIB archives and compare our results with existing results on GSAT, WalkSAT, LSDL, and Grasp. The results demonstrate that DPM with trap avoidance is robust as well as effective for solving hard SAT problems.