A linear-time transformation of linear inequalities into conjunctive normal form
Information Processing Letters
A machine program for theorem-proving
Communications of the ACM
Algorithms for maximum satisfiability using unsatisfiable cores
Proceedings of the conference on Design, automation and test in Europe
Sorting networks and their applications
AFIPS '68 (Spring) Proceedings of the April 30--May 2, 1968, spring joint computer conference
New Encodings of Pseudo-Boolean Constraints into CNF
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Towards robust CNF encodings of cardinality constraints
CP'07 Proceedings of the 13th international conference on Principles and practice of constraint programming
Effective preprocessing in SAT through variable and clause elimination
SAT'05 Proceedings of the 8th international conference on Theory and Applications of Satisfiability Testing
Parallel search for maximum satisfiability
AI Communications - 18th RCRA International Workshop on “Experimental evaluation of algorithms for solving problems with combinatorial explosion”
A cardinality solver: more expressive constraints for free
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
SAT and SMT are still resolution: questions and challenges
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
Conflict directed lazy decomposition
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Compiling finite domain constraints to sat with bee*
Theory and Practice of Logic Programming
Boolean satisfiability for sequence mining
Proceedings of the 22nd ACM international conference on Conference on information & knowledge management
Boolean equi-propagation for concise and efficient SAT encodings of combinatorial problems
Journal of Artificial Intelligence Research
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We introduce Cardinality Networks, a new CNF encoding of cardinality constraints. It improves upon the previously existing encodings such as the sorting networks of Eén and Sörensson (JSAT 2:1---26, 2006) in that it requires much less clauses and auxiliary variables, while arc consistency is still preserved: e.g., for a constraint x 1驴+驴...驴+驴x n 驴驴驴k, as soon as k variables among the x i 's become true, unit propagation sets all other x i 's to false. Our encoding also still admits incremental strengthening: this constraint for any smaller k is obtained without adding any new clauses, by setting a single variable to false. Here we give precise recursive definitions of the clause sets that are needed and give detailed proofs of the required properties. We demonstrate the practical impact of this new encoding by careful experiments comparing it with previous encodings on real-world instances.