Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI-20) Extended Conjunctive Normal Form and An Efficient Algorithm for Cardinality Constraints Zhendong Lei1;2 , Shaowei Cai1;2∗ and Chuan Luo3 1State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, China 2School of Computer Science and Technology, University of Chinese Academy of Sciences, China 3Microsoft Research, China fleizd,
[email protected] Abstract efficient PMS solvers, including SAT-based solvers [Naro- dytska and Bacchus, 2014; Martins et al., 2014; Berg et Satisfiability (SAT) and Maximum Satisfiability al., 2019; Nadel, 2019; Joshi et al., 2019; Demirovic and (MaxSAT) are two basic and important constraint Stuckey, 2019] and local search solvers [Cai et al., 2016; problems with many important applications. SAT Luo et al., 2017; Cai et al., 2017; Lei and Cai, 2018; and MaxSAT are expressed in CNF, which is dif- Guerreiro et al., 2019]. ficult to deal with cardinality constraints. In this One of the most important drawbacks of these logical lan- paper, we introduce Extended Conjunctive Normal guages is the difficulty to deal with cardinality constraints. Form (ECNF), which expresses cardinality con- Indeed, cardinality constraints arise frequently in the encod- straints straightforward and does not need auxiliary ing of many real world situations such as scheduling, logic variables or clauses. Then, we develop a simple and synthesis or verification, product configuration and data min- efficient local search solver LS-ECNF with a well ing. For the above reasons, many works have been done on designed scoring function under ECNF. We also de- finding an efficient encoding of cardinality constraints in CNF velop a generalized Unit Propagation (UP) based formulas [Sinz, 2005; As´ın et al., 2009; Hattad et al., 2017; algorithm to generate the initial solution for local Boudane et al., 2018; Karpinski and Piotrow,´ 2019].