The Hairy Ball Problem is PPAD-Complete

Authors Paul W. Goldberg , Alexandros Hollender

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Paul W. Goldberg
  • Department of Computer Science, University of Oxford, United Kingdom
Alexandros Hollender
  • Department of Computer Science, University of Oxford, United Kingdom

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Paul W. Goldberg and Alexandros Hollender. The Hairy Ball Problem is PPAD-Complete. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem. In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of End-of-Line, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source End-of-Line problem. If the domain is changed to be the torus of genus g >= 2 instead (where the Hairy Ball Theorem also holds), then the problem reduces to a 2(g-1)-source End-of-Line problem. These multiple-source End-of-Line results 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Computational Complexity
  • TFNP
  • PPAD
  • End-of-Line


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