Efficient Encodings of Conditional Cardinality Constraints

TL;DR

This paper introduces new methods for encoding conditional cardinality constraints in propositional logic, improving the efficiency and consistency of SAT solving in real-world applications like pattern mining.

Contribution

It extends existing cardinality constraint encodings to handle conditional constraints while maintaining generalized arc consistency, validated through experimental results.

Findings

Most encodings lose arc consistency when augmented with new variables.

Proposed extensions recover arc consistency in these encodings.

Experimental validation shows improved performance in pattern mining tasks.

Abstract

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 formulation. To avoid combinatorial explosion, 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 $\Phi$ is taken into account leading to conditional constraints of the form $y\rightarrow \Phi$, or $\Phi\rightarrow y$, 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 is a cornerstone, showing the relevance of our proposed approach.