Graph-Based Algorithms for Boolean Function Manipulation
IEEE Transactions on Computers
A linear-time transformation of linear inequalities into conjunctive normal form
Information Processing Letters
On the Size of Ordered Binary Decision Diagrams Representing Threshold Functions
ISAAC '94 Proceedings of the 5th International Symposium on Algorithms and Computation
Generic ILP versus specialized 0-1 ILP: an update
Proceedings of the 2002 IEEE/ACM international conference on Computer-aided design
On Boolean Functions Encodable as a Single Linear Pseudo-Boolean Constraint
CPAIOR '07 Proceedings of the 4th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
CAV '08 Proceedings of the 20th international conference on Computer Aided Verification
New Encodings of Pseudo-Boolean Constraints into CNF
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Circuit complexity and decompositions of global constraints
IJCAI'09 Proceedings of the 21st international jont conference on Artifical intelligence
Construction of efficient BDDs for bounded arithmetic constraints
TACAS'03 Proceedings of the 9th international conference on Tools and algorithms for the construction and analysis of systems
Why cumulative decomposition is not as bad as it sounds
CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
BDDs for pseudo-boolean constraints: revisited
SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
Improving unsatisfiability-based algorithms for boolean optimization
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
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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.