Smart Cubing for Graph Search: A Comparative Study

Propositional satisfiability (SAT) solvers based on conflict-driven clause learning can solve huge instances with millions of variables and clauses [Fichte et al., 2023a]. However, for hard instances, particularly in combinatorial problems, parallelization becomes necessary. The cube-and-conquer technique has proven highly effective for such problems, most notably in resolving the Pythagorean triples conjecture [Heule et al., 2016]. In cube-and-conquer, a look-ahead solver first partitions the search space into disjoint subproblems via cubes (partial assignments), which are then solved independently by CDCL solvers. This independence enables efficient parallel solving. When encoding combinatorial problems into SAT, particularly those involving graphs, we often encounter highly symmetric search spaces. Many mutually isomorphic graphs satisfy the same constraints, but a solver needs to check only one representative, the canonical element, from each isomorphism class. Standard CDCL solvers cannot leverage these symmetries, and static symmetry breaking methods cannot break all symmetries [Codish et al., 2019]. SAT Modulo Symmetries (SMS) [Kirchweger and Szeider, 2021; Kirchweger and Szeider, 2024] addresses this limitation through dynamic symmetry breaking, using a custom propagator that learns symmetry-breaking predicates during the search.