While traditional data mining techniques have been used extensively for finding patterns in databases, they are not always suitable for incorporating user-specified constraints. To overcome this issue, CP and SAT based frameworks for modeling and solving pattern mining tasks have gained a considerable audience in recent years. However, a bottleneck for all these CP and SAT-based approaches is the encoding size which makes these algorithms inefficient for large databases. This paper introduces a practical SAT-based approach to discover efficiently (minimal non-redundant) association rules. First, we present a decomposition-based paradigm that splits the original transaction database into smaller and independent subsets. Then, we show that without producing too large formulas, our decomposition method allows independent mining evaluation on a multi-core machine, improving performance. Finally, an experimental evaluation shows that our method is fast and scale well compared with the existing CP approach even in the sequential case, while significantly reducing the gap with the best state-of-the-art specialized algorithm.

In the encoding of many real-world problems to propositional satisfiability, the cardinality constraint is a recurrent constraint that needs to be managed effectively. Several efficient encodings have been proposed while missing that such a constraint can be involved in a more general propositional formula. To avoid combinatorial explosion, the Tseitin principle usually used to translate such general propositional formula to Conjunctive Normal Form (CNF), introduces fresh propositional variables to represent sub-formulas and/or complex contraints. Thanks to Plaisted and Greenbaum improvement, the polarity of the sub-formula Φ is taken into account leading to conditional constraints of the form y → Φ, or Φ → y, where y is a fresh propositional variable. In the case where Φ represents a cardinality constraint, such translation leads to conditional cardinality constraints subject of the present paper. We first show that when all the clauses encoding the cardinality constraint are augmented with an additional new variable, most of the well-known encodings cease to maintain the generalized arc-consistency property. Then, we consider some of these encodings and show how they can be extended to recover such important property. An experimental validation is conducted on a SAT-based pattern mining application, where such conditional cardinality constraints are a cornerstone, showing the relevance of our proposed approach.