procedure DLM - 98 - BASIC - SAT

In this paper, we present eecient trap-escaping strategies in a search based on discrete Lagrange multipli-ers to solve diicult SAT problems. Although a basic discrete Lagrangian method (DLM) can solve most of the satissable DIMACS SAT benchmarks eeciently, a few of the large benchmarks have eluded solutions by any local-search methods today. These diicult benchmarks generally have many traps that attract local-search trajectories. To this end, we identify the existence of traps when any change to a variable will cause the resulting Lagrangian value to increase. Using the hanoi4 and par16-1 benchmarks, we illustrate that some unsatissed clauses are trapped more often than others. Since it is too diicult to remember explicitly all the traps encountered, we propose to remember these traps implicitly by giving larger increases to Lagrange multipliers of unsatissed clauses that are trapped more often. We illustrate the merit of this new update strategy by solving some of most diicult but satissable SAT benchmarks in the DIMACS archive (hanoi4, hanoi4-simple, par16-1 to par16-5, f2000, and par32-1-c to par32-3-c). Finally, we apply the same algorithm to improve on the solutions of some benchmark MAX-SAT problems that we solved before.