Structural Relaxations by Variable Renaming and Their Compilation for Solving MinCostSAT

Searching for optimal solutions to a problem using lower bounds obtained from a relaxation is a common idea in Heuristic Search and Planning. In SAT and CSPs, however, explicit relaxations are seldom used. In this work, we consider the use of explicit relaxations for solving MinCostSAT, the problem of finding a minimum cost satisfying assignment. We start with the observation that while a number of SAT and CSP tasks have a complexity that is exponential in the treewidth, such models can be relaxed into weaker models of bounded treewidth by a simple form of variable renaming. The relaxed models can then be compiled in polynomial time and space, so that their solutions can be used effectively for pruning the search in the original problem. We have implemented a MinCostSAT solver using this idea on top of two off-the-shelf tools, a d-DNNF compiler that deals with the relaxation, and a SAT solver that deals with the search. The results over the entire suite of 559 problems from the 2006 Weighted Max-SAT Competition are encouraging: SR(w), the new solver, solves 56% of the problems when the bound on the relaxation treewidth is set to w = 8, while Toolbar, the winner, solves 73% of the problems, Lazy, the runner up, 55%, and MinCostChaff, a recent MinCostSAT solver, 26%. The relation between the proposed relaxation method and existing structural solution methods such as cutset decomposition and derivatives such as mini-buckets is also discussed.