Pseudo-Boolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a propositional formula. These BDD-based approaches have some important advantages, such as not being dependent on the size of the coefficients, or being able to share the same BDD for representing many constraints. We first focus on the size of the resulting BDDs, which was considered to be an open problem in our research community. We report on previous work where it was proved that there are Pseudo-Boolean constraints for which no polynomial BDD exists. We also give an alternative and simpler proof assuming that NP is different from Co-NP. More interestingly, here we also show how to overcome the possible exponential blowup of BDDs by coefficient decomposition. This allows us to give the first polynomial generalized arc-consistent ROBDD-based encoding for Pseudo-Boolean constraints. Finally, we focus on practical issues: we show how to efficiently construct such ROBDDs, how to encode them into SAT with only 2 clauses per node, and present experimental results that confirm that our approach is competitive with other encodings and state-of-the-art Pseudo-Boolean solvers.
Abstract. We improve the state of the art for solving car-sequencing problems by combining together the strengths of SAT and CP. We compare both pure SAT and hybrid CP/SAT models. Three features of these models are crucial to success. For quickly finding solutions, advanced CP heuristics are important and good propagation (either by a specialized propagator or by a sophisticated SAT encoding that simulates one) is necessary. For proving infeasibility, clause learning in the SAT solver is critical. Our models contain a number of other novelties. In our hybrid models, we develop a novel linear time mechanism for explaining failure and pruning for the ATMOSTSEQCARD constraint. In our SAT models, we describe powerful encodings for the same constraint. Our study demonstrates some of the potential and complementarity of SAT and hybrid methods for solving complex constraint models.
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