模$p$变元的拓扑刻画及其对项链分裂的启示
A Topological Characterization of Modulo-$p$ Arguments and Implications for Necklace Splitting
摘要：
类PPA- $p$最近引起了人们的注意，因为它们是主要的候选者，用于捕获使用$p$窃贼的项链分裂的复杂性，对于素数$p$。然而，这些类并没有已知的拓扑性质的完整问题，这阻碍了解决项链分裂问题的复杂性的任何进展。相反，拓扑问题在获得PPAD和PPA的完备性结果中是关键的，例如找到纳什均衡的PPAD完备性[Daskalakis等人，2009，Chen等人，2009 b]和2个小偷的项链分裂的PPA完全性[Filos-Ratsikas和Goldberg，2019]。本文给出了类PPA- $p$的第一个拓扑刻画.首先，我们证明了与Tucker引理的一个简单推广相关的计算问题，称为$p$ -B-Tucker，以及相关的Borsuk-Ulam型定理，$p$ -B-Borsuk-Ulam，是PPA- $p$ -完全的。然后，我们证明了著名的BSS定理的计算版本[Barany等人，1981]，以及相关的BSS塔克问题是PPA- $p$ -完全的。最后，利用Tucker引理的一个不同的推广（称为$\mathbb{Z}_p$ -star-Tucker），我们证明了它是PPA- $p$ -完全的，我们证明了$p$ -小偷项链分裂是PPA- $p$ .后一个结果给出了项链分裂定理的一个新的组合证明，这是除了Meunier [2014]之外唯一的证明。我们所有的包容性结果是通过一个新的组合证明$\mathbb{Z}_p$ -版本的塔克引理，是一个自然推广的标准组合证明塔克引理弗罗因德和托德[1981]。我们相信，这种新的证明技术是独立的利益。
英文摘要：
The classes PPA-$p$ have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with $p$ thieves, for prime $p$. However, these classes were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [Daskalakis et al., 2009, Chen et al., 2009b] and the PPA-completeness of Necklace Splitting with 2 thieves [Filos-Ratsikas and Goldberg, 2019]. In this paper, we provide the first topological characterization of the classes PPA-$p$. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed $p$-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, $p$-polygon-Borsuk-Ulam, are PPA-$p$-complete. Then, we show that the computational version of the well-known BSS Theorem [Barany et al., 1981], as well as the associated BSS-Tucker problem are PPA-$p$-complete. Finally, using a different generalization of Tucker's Lemma (termed $\mathbb{Z}_p$-star-Tucker), which we prove to be PPA-$p$-complete, we prove that $p$-thief Necklace Splitting is in PPA-$p$. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [2014]. All of our containment results are obtained through a new combinatorial proof for $\mathbb{Z}_p$-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [1981]. We believe that this new proof technique is of independent interest.
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