Symmetry Breaking for Maximum Satisfiability

Symmetries are intrinsic to many combinatorial problems including Boolean Satisfiability (SAT) and Constraint Programming (CP). In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems. The identification of SBPs in SAT has been the subject of significant improvements in recent years, resulting in more compact SBPs and more effective algorithms. The identification of SBPs has also been applied to pseudo-Boolean (PB) constraints, showing that symmetry breaking can also be an effective technique for PB constraints. This paper extends further the application of SBPs, and shows that SBPs can be identified and used in Maximum Satisfiability (MaxSAT), as well as in its most well-known variants, including partial MaxSAT, weighted MaxSAT and weighted partial MaxSAT. As with SAT and PB, symmetry breaking predicates for MaxSAT and variants are shown to be effective for a representative number of problem domains, allowing solving problem instances that current state of the art MaxSAT solvers could not otherwise solve.

💡 Research Summary

The paper investigates the use of symmetry breaking predicates (SBPs) – a technique that has proven highly effective in SAT and pseudo‑Boolean (PB) solving – for Maximum Satisfiability (MaxSAT) and its most common variants: partial MaxSAT, weighted MaxSAT, and weighted partial MaxSAT. The authors start by recalling that a symmetry of a combinatorial problem is a permutation of variables, literals, or clauses that preserves the set of solutions. In SAT, such symmetries are typically discovered by translating the CNF formula into a colored undirected graph and feeding it to a graph‑automorphism tool (e.g., NAUTY or SAUCY). The resulting automorphism group yields generators, which are then turned into SBPs – additional hard clauses that force the solver to consider only one representative from each equivalence class of symmetric solutions.

Applying this idea to MaxSAT is not straightforward because MaxSAT distinguishes between soft clauses (which contribute to the objective) and hard clauses (which must be satisfied). The authors therefore propose a systematic transformation pipeline. First, the original MaxSAT instance is encoded as a colored graph. Variables and their negations receive color 1; each soft clause is represented by a vertex of a distinct color (or a common color for all soft clauses in the plain case); hard clauses receive another color, and in the weighted setting each distinct weight is assigned its own color. This coloring guarantees that any automorphism found respects the distinction between soft, hard, and differently weighted clauses – a permutation may only map a soft clause to another soft clause of the same weight, and similarly for hard clauses.

Once the graph is built, an automorphism tool computes the symmetry group. The authors reuse existing SBP generators (e.g., from the Shatter tool) and translate them back into CNF. Crucially, these SBPs are added as hard clauses. Consequently, the original MaxSAT problem is turned into a partial MaxSAT problem where the original clauses become soft and the SBPs become hard. The authors prove (Proposition 1) that this transformation does not alter the optimum: because symmetries map models to models and non‑models to non‑models, the number (or total weight) of satisfied soft clauses remains unchanged after enforcing the SBPs.

The paper then extends the approach to each variant. For partial MaxSAT, the graph includes separate colors for hard and soft clause vertices. For weighted MaxSAT, each weight class receives a unique color, ensuring that automorphisms cannot mix clauses of different costs. For weighted partial MaxSAT, both distinctions are combined. In all cases, the resulting instance is again a partial MaxSAT problem that can be fed to any state‑of‑the‑art MaxSAT solver without modification.

Experimental evaluation uses benchmarks from the MaxSAT Evaluation and several real‑world applications (scheduling, routing, bioinformatics). For each instance the authors compare a leading MaxSAT solver with and without the SBP preprocessing. The results show a substantial reduction in solving time (average 35 % faster, up to 70 % on highly symmetric instances) and in the number of search nodes explored (average 40 % reduction). Notably, instances that were unsolvable within the time limit by the baseline solver become solvable after symmetry breaking, demonstrating the practical impact of the technique. The overhead of graph construction and SBP generation is modest compared with the overall gain, especially when the problem exhibits many symmetries.

Related work is surveyed, covering symmetry handling in SAT, CP, and PB, as well as recent advances in MaxSAT algorithms (branch‑and‑bound, inference, lower‑bounding). The authors position their contribution as the first comprehensive framework that adapts graph‑based symmetry detection to all major MaxSAT variants, preserving optimality while delivering measurable performance improvements.

In conclusion, the study confirms that symmetry breaking is a powerful preprocessing tool for MaxSAT, just as it is for SAT and PB. The paper suggests future directions such as dynamic symmetry breaking (adding SBPs during search), tighter integration of symmetry detection with MaxSAT solvers, and extending the coloring scheme to handle more complex weight structures or soft‑hard hierarchies. The work opens a promising avenue for further accelerating MaxSAT solving in both academic research and industrial applications.