Towards an Optimal CNF Encoding of Boolean Cardinality Constraints

We consider the problem of encoding Boolean cardinality constraints in conjunctive normal form (CNF). Boolean cardinality constraints are formulae expressing that at most (resp. at least) k out of n propositional variables or formulae are true. We present a unifying framework for a whole family of such encodings encompassing previously proposed solutions. We give two novel encodings that improve upon existing results, one which requires only 7n clauses and 2n additional variables, and another one demanding O(n · k) clauses, but with the advantage that inconsistencies can be detected in linear time by unit propagation alone. Moreover, we prove a linear lower bound on the number of required clauses for any such encoding.