License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.65
URN: urn:nbn:de:0030-drops-106416
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10641/
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Goldberg, Paul W. ; Hollender, Alexandros

The Hairy Ball Problem is PPAD-Complete

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LIPIcs-ICALP-2019-65.pdf (0.5 MB)

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 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.

BibTeX - Entry

@InProceedings{goldberg_et_al:LIPIcs:2019:10641,
  author =	{Paul W. Goldberg and Alexandros Hollender},
  title =	{{The Hairy Ball Problem is PPAD-Complete}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{65:1--65:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10641},
  URN =		{urn:nbn:de:0030-drops-106416},
  doi =		{10.4230/LIPIcs.ICALP.2019.65},
  annote =	{Keywords: Computational Complexity, TFNP, PPAD, End-of-Line}
}

Keywords: Computational Complexity, TFNP, PPAD, End-of-Line
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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