IntSat: Integer Linear Programming by Conflict-Driven Constraint-Learning

State-of-the-art SAT solvers are nowadays able to handle huge real-world instances. The key to this success is the so-called Conflict-Driven Clause-Learning (CDCL) scheme, which encompasses a number of techniques that exploit the conflicts that are encountered during the search for a solution. In this article we extend these techniques to Integer Linear Programming (ILP), where variables may take general integer values instead of purely binary ones, constraints are more expressive than just propositional clauses, and there may be an objective function to optimise. We explain how these methods can be implemented efficiently, and discuss possible improvements. Our work is backed with a basic implementation that shows that, even in this far less mature stage, our techniques are already a useful complement to the state of the art in ILP solving.

A New Look at BDDs for Pseudo-Boolean Constraints

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 phcoefficient 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.