When solving a combinatorial problem using propositional satisfiability (SAT), the encoding of the problem is of vital importance. We study encodings of Pseudo-Boolean (PB) constraints, a common type of arithmetic constraint that appears in a wide variety of combinatorial problems such as timetabling, scheduling, and resource allocation. In some cases PB constraints occur together with at-most-one (AMO) constraints over subsets of their variables (forming PB(AMO) constraints). Recent work has shown that taking account of AMOs when encoding PB constraints using decision diagrams can produce a dramatic improvement in solver efficiency. In this paper we extend the approach to other state-of-the-art encodings of PB constraints, developing several new encodings for PB(AMO) constraints. Also, we present a more compact and efficient version of the popular Generalized Totalizer encoding, named Reduced Generalized Totalizer. This new encoding is also adapted for PB(AMO) constraints for a further gain. Our experiments show that the encodings of PB(AMO) constraints can be substantially smaller than those of PB constraints. PB(AMO) encodings allow many more instances to be solved within a time limit, and solving time is improved by more than one order of magnitude in some cases. We also observed that there is no single overall winner among the considered encodings, but efficiency of each encoding may depend on PB(AMO) characteristics such as the magnitude of coefficient values.

在使用命题可满足性 (SAT) 解决组合问题时，问题的编码至关重要。我们研究伪布尔 (PB) 约束的编码，这是一种常见的算术约束类型，出现在各种组合问题中，例如时间表、调度和资源分配。在某些情况下，PB 约束与其变量子集上的至多一个 (AMO) 约束一起出现（形成 PB(AMO) 约束）。最近的工作表明，在使用决策图对 PB 约束进行编码时考虑 AMO 可以显着提高求解器效率。在本文中，我们将方法扩展到 PB 约束的其他最先进的编码，为 PB(AMO) 约束开发了几种新的编码。还，我们提出了流行的广义累加器编码的更紧凑和更高效的版本，称为简化广义累加器。这种新编码也适用于 PB(AMO) 约束以获得进一步的增益。我们的实验表明，PB(AMO) 约束的编码可以大大小于 PB 约束的编码。PB(AMO) 编码允许在时间限制内求解更多实例，并且在某些情况下求解时间提高了不止一个数量级。我们还观察到，在所考虑的编码中没有单一的整体赢家，但每种编码的效率可能取决于 PB(AMO) 特性，例如系数值的大小。我们的实验表明，PB(AMO) 约束的编码可以大大小于 PB 约束的编码。PB(AMO) 编码允许在时间限制内求解更多实例，并且在某些情况下求解时间提高了不止一个数量级。我们还观察到，在所考虑的编码中没有单一的整体赢家，但每种编码的效率可能取决于 PB(AMO) 特性，例如系数值的大小。我们的实验表明，PB(AMO) 约束的编码可以大大小于 PB 约束的编码。PB(AMO) 编码允许在时间限制内求解更多实例，并且在某些情况下求解时间提高了不止一个数量级。我们还观察到，在所考虑的编码中没有单一的整体赢家，但每种编码的效率可能取决于 PB(AMO) 特性，例如系数值的大小。