A New Look at BDDs for Pseudo-Boolean Constraints

Abstract

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.

Generalized Totalizer Encoding for Pseudo-Boolean Constraints

“…We believe that such a localized reasoning could be a cause of relatively better performance of GTE as reported in Section 4. It is worth noting that the worst case scenario for GTE, when weights are of the form a i , where a ≥ 2, would generate a polynomial size formula for BDD based approaches [1,6,11].…”

“…Similar resource limitations were used during the last pseudo-Boolean (PB) evaluation of 2012 3 . For a fair comparison, we implemented GTE (gte) in the PBLib [29] (version 1.2) open source library which contains a plethora of encodings, namely, Adder Networks (adder) [11,30], Sorting Networks (sorter) [11], watchdog (watchdog) [7], Binary Merger (bin-merger) [20], Sequential Weighted Counter (swc) [14], and BDDs (bdd) [1]. A new encoding in PBLib can be added by implementing encode method of the base class Encoder.…”

Encoding Linear Constraints into SAT

“…In the first transformation, domain consistency is lost. Notice, however, that a consistent propagator for linear equality would solve the NP-complete problem subset sum (see (Abío et al, 2012)), so, in principle, there is no consistent propagator for equality constraints that runs in polynomial time (unless P = NP).…”

“…Several improvements were made by Abío et al (2012). First, a faster algorithm for constructing the reduced BDD is presented.…”

“…Secondly, BDDs are one of the best methods for encoding pseudo-Boolean constraints into SAT by Abío et al (2012), and MDDs seems the natural tool to generalize the pseudo-Boolean encoding.…”