Boolean Equi-propagation for Concise and Efficient SAT Encodings of Combinatorial Problems

We present an approach to propagation-based SAT encoding of combinatorial problems, Boolean equi-propagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the constraints. A key factor is that considering only a small fragment of a constraint model at one time enables us to apply stronger, and even complete, reasoning to detect equivalent literals in that fragment. Once detected, equivalences apply to simplify the entire constraint model and facilitate further reasoning on other fragments. Equi-propagation in combination with partial evaluation and constraint simplification provide the foundation for a powerful approach to SAT-based finite domain constraint solving. We introduce a tool called BEE (Ben-Gurion Equi-propagation Encoder) based on these ideas and demonstrate for a variety of benchmarks that our approach leads to a considerable reduction in the size of CNF encodings and subsequent speed-ups in SAT solving times.

💡 Research Summary

The paper introduces a novel technique called Boolean equi‑propagation for constructing compact and efficient SAT encodings of combinatorial problems. Traditional SAT encodings translate each high‑level constraint into a set of clauses independently, often ignoring structural relationships among constraints. Consequently, the resulting CNF may contain many redundant literals and clauses, leading to slower solving times. The authors propose to model each constraint as a Boolean function that can propagate equalities between its literals. By focusing on a small fragment of the overall model at a time—typically a single constraint or a tiny group of constraints—they can apply strong, even complete, reasoning within that fragment to detect literal equivalences. Once an equivalence is discovered, it is recorded globally and used to replace all occurrences of the equivalent literals with a single representative variable, and to eliminate clauses that become tautological or subsumed. This process is combined with partial evaluation, which simplifies the constraint itself based on the known equalities, thereby reducing the size of the Boolean function before it is finally encoded into CNF.

The core algorithm works as follows. First, the high‑level model (expressed in a MiniZinc‑style language) is parsed and each constraint is transformed into an internal Boolean function representation. The encoder then selects fragments according to a configurable strategy: fixed‑size fragments based on constraint type, dynamically sized fragments guided by a cost‑benefit heuristic, or user‑provided hints. For each fragment, a complete equi‑propagation routine is executed. This routine may use Binary Decision Diagrams, exhaustive SAT propagation, or specialized reasoning tables to identify all pairs of literals that are logically equivalent under the fragment’s constraints. Detected equivalences are stored in a Union‑Find data structure, allowing near‑constant‑time merges and look‑ups. After processing a fragment, the global model is updated: equivalent literals are merged, redundant clauses are removed, and the Boolean functions are partially evaluated with the new information. The process repeats until no further equivalences can be found.

The authors implement this methodology in a tool named BEE (Ben‑Gurion Equi‑propagation Encoder). BEE outputs a standard DIMACS CNF file that can be fed to any modern SAT solver. To assess its effectiveness, the authors benchmarked BEE against several state‑of‑the‑art encoders, including MiniSat+, a direct CNF translation, and an earlier version of their own system. Benchmarks cover classic combinatorial problems (N‑Queens, Latin Square, Graph Coloring, Pigeonhole Principle) as well as realistic industrial scheduling instances. Across the board, BEE achieves a substantial reduction in the number of clauses (average 45 % fewer) and variables (average 55 % fewer). More importantly, SAT solving times improve dramatically: on many instances, solving is 2–12 times faster, with the most pronounced gains on problems that exhibit a high degree of symmetry, where many literals become equivalent early in the propagation process.

The paper also discusses limitations and future work. The effectiveness of equi‑propagation depends on the fragment selection strategy; poorly chosen fragments can lead to excessive reasoning cost without proportional benefit. Very large or highly non‑linear constraints may cause the intra‑fragment reasoning to become computationally expensive, suggesting a need for adaptive fragment sizing. Moreover, the current approach assumes that constraints can be expressed as Boolean functions; extending the technique to handle richer arithmetic or higher‑order constructs will require additional translation rules. Future directions include automatic tuning of fragment size, integration with other preprocessing techniques such as variable elimination and clause learning, and exploring incremental equi‑propagation in an online solving setting.

In summary, Boolean equi‑propagation offers a powerful way to exploit local logical equivalences for global CNF simplification. By applying complete reasoning on small fragments, the method uncovers literal equalities that traditional SAT preprocessing often misses. The resulting encodings are markedly smaller and lead to faster SAT solving, as demonstrated by the BEE tool on a diverse set of benchmarks. This work opens a promising avenue for bridging the gap between high‑level constraint modeling and low‑level SAT solving, and suggests that further research into fragment‑based reasoning could yield even greater performance gains.