Integer and combinatorial optimization
Integer and combinatorial optimization
Counting solutions to Presburger formulas: how and why
PLDI '94 Proceedings of the ACM SIGPLAN 1994 conference on Programming language design and implementation
A linear-time transformation of linear inequalities into conjunctive normal form
Information Processing Letters
Counting Models Using Connected Components
Proceedings of the Seventeenth National Conference on Artificial Intelligence and Twelfth Conference on Innovative Applications of Artificial Intelligence
Applying SAT Methods in Unbounded Symbolic Model Checking
CAV '02 Proceedings of the 14th International Conference on Computer Aided Verification
Generic ILP versus specialized 0-1 ILP: an update
Proceedings of the 2002 IEEE/ACM international conference on Computer-aided design
A fast pseudo-boolean constraint solver
Proceedings of the 40th annual Design Automation Conference
Computing Prime Implicants by Integer Programming
ICTAI '96 Proceedings of the 8th International Conference on Tools with Artificial Intelligence
Algorithms and Complexity Results for #SAT and Bayesian Inference
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Counting the solutions of Presburger equations without enumerating them
Theoretical Computer Science - Implementation and application automata
Counting CSP solutions using generalized XOR constraints
AAAI'07 Proceedings of the 22nd national conference on Artificial intelligence - Volume 1
Counting solutions of integer programs using unrestricted subtree detection
CPAIOR'08 Proceedings of the 5th international conference on Integration of AI and OR techniques in constraint programming for combinatorial optimization problems
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This paper addresses the problem of counting models in integer linear programming (ILP) using Boolean Satisfiability (SAT) techniques, and proposes two approaches to solve this problem. The first approach consists of encoding ILP instances into pseudo-Boolean (PB) instances. Moreover, the paper introduces a model counter for PB constraints, which can be used for counting models in PB as well as in ILP. A second alternative approach consists of encoding instances of ILP into instances of SAT. A two-step procedure is proposed, consisting of first mapping the ILP instance into PB constraints and then encoding the PB constraints into SAT. One key observation is that not all existing PB to SAT encodings can be used for counting models. The paper provides conditions for PB to SAT encodings that can be safely used for model counting, and proves that some of the existing encodings are safe for model counting while others are not. Finally, the paper provides experimental results, comparing the PB and SAT approaches, as well as existing alternative solutions.