Extended Conjunctive Normal Form and An Efficient Algorithm for Cardinality Constraints

Abstract

Satisfiability (SAT) and Maximum Satisfiability (MaxSAT) are two basic and important constraint problems with many important applications. SAT and MaxSAT are expressed in CNF, which is difficult to deal with cardinality constraints. In this paper, we introduce Extended Conjunctive Normal Form (ECNF), which expresses cardinality constraints straightforward and does not need auxiliary variables or clauses. Then, we develop a simple and efficient local search solver LS-ECNF with a well designed scoring function under ECNF. We also develop a generalized Unit Propagation (UP) based algorithm to generate the initial solution for local search. We encode instances from Nurse Rostering and Discrete Tomography Problems into CNF with three different cardinality constraint encodings and ECNF respectively. Experimental results show that LS-ECNF has much better performance than state of the art MaxSAT, SAT, Pseudo-Boolean and ILP solvers, which indicates solving cardinality constraints with ECNF is promising.