A Pigeon-Hole Based Encoding of Cardinality Constraints

In this paper, we propose a new encoding of the cardinality constraint ∑i=1 xi > b. It makes an original use of the general formulation of the Pigeon-Hole principle to derive a formula in conjunctive normal form (CNF). Our Pigeon-Hole based CNF encoding can be seen as a nice and simple way to express the semantic of the cardinality constraint, that can be defined as how to put b pigeons into n holes. To derive an efficient CNF encoding that ensures constraint propagation, we exploit the set of symmetries of the Pigeon-Hole based formulation to derive an efficient CNF encoding of the cardinality constraint. More interestingly, the final CNF formula contains b× (n− b) variables and clauses and belongs to the wellknown Reverse-Horn tractable CNF formula, which can be decided by unit propagation. Our proposed Pigeon-Hole based encoding is theoretically compared with the currently well-known CNF encoding of the cardinality constraint.