Faster Certified Symmetry Breaking Using Orders With Auxiliary Variables

TL;DR

This paper introduces a new method for symmetry breaking in combinatorial solving that uses auxiliary variables to encode orders, significantly improving speed and efficiency in proof logging and checking.

Contribution

It proposes a novel approach to encode orders with auxiliary variables, overcoming limitations of previous methods relying on big integers.

Findings

Order encoding with auxiliary variables yields significant speed-ups.

Method improves proof logging and checking efficiency.

Experimental results demonstrate practical benefits in SAT symmetry breaking.

Abstract

Symmetry breaking is a crucial technique in modern combinatorial solving, but it is difficult to be sure it is implemented correctly. The most successful approach to deal with bugs is to make solvers certifying, so that they output not just a solution, but also a mathematical proof of correctness in a standard format, which can then be checked by a formally verified checker. This requires justifying symmetry reasoning within the proof, but developing efficient methods for this has remained a long-standing open challenge. A fully general approach was recently proposed by Bogaerts et al. (2023), but it relies on encoding lexicographic orders with big integers, which quickly becomes infeasible for large symmetries. In this work, we develop a method for instead encoding orders with auxiliary variables. We show that this leads to orders-of-magnitude speed-ups in both theory and practice by running experiments on proof logging and checking for SAT symmetry breaking using the state-of-the-art satsuma symmetry breaker and the VeriPB proof checking toolchain.