Orbitopal Fixing in SAT

2026-01-23

AI Summary

Boolean satisfiability (SAT) solvers often redundantly explore equivalent regions of the search space due to symmetries. This work proposes a static symmetry-breaking method based on orbitopal fixing, adapted for the first time to proof-logging SAT solvers that support formal verification. By introducing only unit clauses, the approach efficiently handles symmetries without interfering with the solver’s native heuristics. It integrates orbitopal fixing techniques from mixed-integer programming with the substitution redundancy proof system. Implemented in the solver satsuma, the method yields significant speedups on benchmarks rich in symmetries while incurring negligible performance overhead on other instances, all while producing concise, formally verifiable proof certificates.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Data transparency and provenance

Abstract

Despite their sophisticated heuristics, boolean satisfiability (SAT) solvers are still vulnerable to symmetry, causing them to visit search regions that are symmetric to ones already explored. While symmetry handling is routine in other solving paradigms, integrating it into state-of-the-art proof-producing SAT solvers is difficult: added reasoning must be fast, non-interfering with solver heuristics, and compatible with formal proof logging. To address these issues, we present a practical static symmetry breaking approach based on orbitopal fixing, a technique adapted from mixed-integer programming. Our approach adds only unit clauses, which minimizes downstream slowdowns, and it emits succinct proof certificates in the substitution redundancy proof system. Implemented in the satsuma tool, our methods deliver consistent speedups on symmetry-rich benchmarks with negligible regressions elsewhere.

Problem

• symmetry
• SAT solvers
• orbitopal fixing
• proof logging
• search space redundancy

Innovation

• orbitopal fixing
• symmetry breaking
• SAT solving
• proof logging
• unit clauses