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Efficient SAT-based encodings of conditional cardinality constraints. (English) Zbl 1415.68185
Barthe, Gilles (ed.) et al., LPAR-22. 22nd international conference on logic for programming, artificial intelligence and reasoning, Awassa, Ethiopia, November 17–21, 2018. Selected papers. Manchester: EasyChair. EPiC Ser. Comput. 57, 181-195 (2018).
Summary: 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 constraints. Thanks to the improvement by D. A. Plaisted and S. Greenbaum [J. Symb. Comput. 2, 293–304 (1986; Zbl 0636.68119)], the polarity of the sub-formula $$\Phi$$ is taken into account leading to conditional constraints of the form $$y\to\Phi\to y$$ or $$\Phi\to y$$, where $$y$$ is a fresh propositional variable. In the case where $$\Phi$$ 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.
For the entire collection see [Zbl 1407.68021].

##### MSC:
 68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
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