毛球问题是PPAD-完全的
The Hairy Ball Problem is PPAD-Complete
摘要：
毛球定理指出，偶数维球面上的每个连续切向量场必须有一个零点。我们证明了相关的计算问题（a）计算近似零是PPAD-完全的，（b）计算精确零是FIXP-困难的。我们还考虑了毛球定理的环形，而不是球形域，并表明近似问题仍然PPAD完全。在概念层面上，我们的PPAD成员的结果是特别有趣的，因为他们在很大程度上依赖于调查的多源变量的END-OF-LINE，规范的PPAD-完全的问题。我们的结果对这些新的终端线的变种是独立的利益，并提供了新的工具，显示成员在PPAD。特别是，我们使用它们来提供第一个完整的证明PPAD的完整性的不平衡问题定义的Beame等人。在1998年。
英文摘要：
The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of END-OF-LINE, the canonical PPAD-complete problem. Our results on these new END-OF-LINE variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the IMBALANCE problem defined by Beame et al. in 1998.
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