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.

[1]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[2]  Rina Dechter,et al.  Constraint Processing , 1995, Lecture Notes in Computer Science.

[3]  Enrico Giunchiglia,et al.  Solving Optimization Problems with DLL , 2006, ECAI.

[4]  Blai Bonet,et al.  Planning as heuristic search , 2001, Artif. Intell..

[5]  Georg Gottlob,et al.  Hypertree decompositions and tractable queries , 1998, PODS '99.

[6]  Javier Larrosa,et al.  New Inference Rules for Efficient Max-SAT Solving , 2006, AAAI.

[7]  Richard E. Korf,et al.  Finding Optimal Solutions to Rubik's Cube Using Pattern Databases , 1997, AAAI/IAAI.

[8]  Rina Dechter,et al.  Enhancement Schemes for Constraint Processing: Backjumping, Learning, and Cutset Decomposition , 1990, Artif. Intell..

[9]  Pierre Marquis,et al.  Compiling propositional weighted bases , 2004, Artif. Intell..

[10]  Thomas Schiex,et al.  Semiring-Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison , 1999, Constraints.

[11]  Felip Manyà,et al.  Detecting Disjoint Inconsistent Subformulas for Computing Lower Bounds for Max-SAT , 2006, AAAI.

[12]  Adnan Darwiche,et al.  Recursive conditioning , 2001, Artif. Intell..

[13]  S. Malik,et al.  Solving the Minimum-Cost Satisfiability Problem Using SAT Based Branch-and-Bound Search , 2006, 2006 IEEE/ACM International Conference on Computer Aided Design.

[14]  Olivier Coudert,et al.  New Ideas for Solving Covering Problems , 1995, 32nd Design Automation Conference.

[15]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[16]  Matthias F. Stallmann,et al.  Optimization algorithms for the minimum-cost satisfiability problem , 2004 .

[17]  Javier Larrosa,et al.  Resolution in Max-SAT and its relation to local consistency in weighted CSPs , 2005, IJCAI.

[18]  Rina Dechter,et al.  Bucket Elimination: A Unifying Framework for Reasoning , 1999, Artif. Intell..

[19]  Sharad Malik,et al.  Conflict driven learning in a quantified Boolean Satisfiability solver , 2002, ICCAD 2002.

[20]  Adnan Darwiche,et al.  New Advances in Compiling CNF into Decomposable Negation Normal Form , 2004, ECAI.

[21]  Rina Dechter,et al.  Mini-Buckets: A General Scheme for Generating Approximations in Automated Reasoning , 1997, IJCAI.

[22]  Pierre Marquis,et al.  A Knowledge Compilation Map , 2002, J. Artif. Intell. Res..

[23]  Simon de Givry,et al.  Solving Max-SAT as Weighted CSP , 2003, CP.

[24]  Teresa Alsinet,et al.  Improved Exact Solvers for Weighted Max-SAT , 2005, SAT.

[25]  Adnan Darwiche,et al.  Node Splitting: A Scheme for Generating Upper Bounds in Bayesian Networks , 2007, UAI.

[26]  Vasco M. Manquinho,et al.  Search pruning techniques in SAT-based branch-and-bound algorithmsfor the binate covering problem , 2002, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[27]  Sharad Malik,et al.  Efficient conflict driven learning in a Boolean satisfiability solver , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[28]  Adnan Darwiche,et al.  Decomposable negation normal form , 2001, JACM.

[29]  Jonathan Schaeffer,et al.  Pattern Databases , 1998, Comput. Intell..

[30]  Rina Dechter,et al.  On Finding Minimal w-cutset , 2004, UAI.

[31]  Alan K. Mackworth.  Consistency in Networks of Relations , 1977, Artif. Intell..

[32]  Francesca Rossi,et al.  Principles and Practice of Constraint Programming – CP 2003 , 2003, Lecture Notes in Computer Science.

[33]  Joao Marques-Silva,et al.  GRASP-A new search algorithm for satisfiability , 1996, Proceedings of International Conference on Computer Aided Design.