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.

毛球定理指出，偶数维球面上的每个连续切向量场都必须为零。我们证明 (a) 计算近似零的相关计算问题是 PPAD 完全的，并且 (b) 计算精确零是 FIXP 困难的。我们还考虑了环形域而不是球形域上的毛球定理，并表明近似问题仍然是 PPAD 完备的。在概念层面上，我们的 PPAD 成员结果特别有趣，因为它们严重依赖于对End-of-Line的多源变体的调查，这是规范的 PPAD 完全问题。我们在这些新的End-of-Line 上的结果变体具有独立的兴趣，并提供了显示 PPAD 成员身份的新工具。特别是，我们使用它们为Beame 等人定义的不平衡问题提供了 PPAD 完整性的第一个完整证明。1998 年。