Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is \(\mathcal {N}\mathcal {P}\)-hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.

中文翻译：

---

MaxSAT伪布尔约束的基于N级模的CNF编码

各个领域中的许多组合问题都可以转化为最大满意度（MaxSAT）问题。尽管一般问题是\（\ mathcal {N} \ mathcal {P} \）-由于致力于有效求解器的大量努力，可能解决越来越困难的实际问题。约束编码的技术与为MaxSAT设计算法的技术一样重要。在本文中，我们介绍了几种基于联合算术形式（CNF）公式的布尔可满足性问题中的伪布尔约束编码方法，这些编码方法基于模块化算术的思想，并且仅为权重的每个唯一组合生成辅助变量。这些技术可以有效地编码和解决MaxSAT问题。特别是，我们的求解器从2010年到2012年赢得了部分MaxSAT工业类，并在2017年MaxSAT评估的主要加权轨道中排名第二。我们证明了编码的正确性和伪多项式空间的复杂性，还给出了启发式的模块化算术基本选择方法。我们的实验结果表明，我们的编码对约束进行了紧凑编码，并且所获得的子句由最新的SAT求解器有效地处理。