Optimal Base Encodings for Pseudo-Boolean Constraints

TL;DR

This paper introduces an algorithm to find optimal base encodings for Pseudo-Boolean constraints, significantly reducing CNF sizes and improving SAT solving efficiency, especially for large numbers up to one million.

Contribution

It formalizes the optimal base problem, presents a scalable algorithm, and demonstrates its effectiveness within the MINISAT+ solver for encoding Pseudo-Boolean constraints.

Findings

Algorithm scales to bases up to 1,000,000

Reduces CNF sizes compared to prime-based encodings

Improves SAT solving times for many instances

Abstract

This paper formalizes the optimal base problem, presents an algorithm to solve it, and describes its application to the encoding of Pseudo-Boolean constraints to SAT. We demonstrate the impact of integrating our algorithm within the Pseudo-Boolean constraint solver MINISAT+. Experimentation indicates that our algorithm scales to bases involving numbers up to 1,000,000, improving on the restriction in MINISAT+ to prime numbers up to 17. We show that, while for many examples primes up to 17 do suffice, encoding with respect to optimal bases reduces the CNF sizes and improves the subsequent SAT solving time for many examples.