The complexity classes PPA-k, $k\ge 2$, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with k thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-k in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.

中文翻译：

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PPA-k 类：存在于参数模 k

复杂度类 PPA- k，$克\ge 2$，最近成为捕获公平划分中重要问题的复杂性的主要候选者，特别是 Alon 的项链分裂问题与k thieves。事实上，对于 PPA = PPA-2，两个小偷的问题已经被证明是完整的。在这项工作中，我们提出了结构结果，为进一步研究这些类提供了坚实的基础。即，我们研究的类PPA- ķ下图灵削减（i）的等效定义的术语，（ⅱ）内的结构，（ⅲ）至彼此和其他TFNP类的关系，和（iv）闭合。