BDDs for Pseudo-Boolean Constraints

Abstract

Pseudo-Boolean constraints are omnipresent in practical applications, and therefore a significant effort has been devoted to the development of good SAT encoding techniques for these constraints. Several of these encodings are based on building Binary Decision Diagrams (BDDs) and translating these into CNF. Indeed, BDD-based encodings have important advantages, such as sharing the same BDD for representing many constraints. Here we first prove that, unless NP = Co-NP, there are Pseudo- Boolean constraints that admit no variable ordering giving a polynomial (Reduced, Ordered) BDD. As far as we know, this result is new (in spite of some misleading information in the literature). It gives several interesting insights, also relating proof complexity and BDDs. But, more interestingly for practice, here we also show how to overcome this theoretical limitation by coefficient decomposition. This allows us to give the first polynomial arc-consistent BDD-based encoding for Pseudo-Boolean constraints.