The Hairy Ball Problem is PPAD-Complete

Paul W. Goldberg; Alexandros Hollender

doi:10.1016/j.jcss.2021.05.004

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arXiv · 1902.07657

The Hairy Ball Problem is PPAD-Complete

Paul W. Goldberg·Alexandros Hollender

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Abstract

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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Paul W. Goldberg, Alexandros Hollender. 2019-02-20. The Hairy Ball Problem is PPAD-Complete. https://doi.org/10.1016/j.jcss.2021.05.004

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