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DOI: 10.4230/LIPIcs.CCC.2022.33

Further Collapses in TFNP

Authors: Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

We show EOPL = PLS ∩ PPAD. Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse CLS = PLS ∩ PPAD by Fearnley et al. (STOC 2021). We also prove a companion result SOPL = PLS ∩ PPADS, where SOPL is the class associated with the Sink-of-Potential-Line problem.

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Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further Collapses in TFNP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goos_et_al:LIPIcs.CCC.2022.33,
  author =	{G\"{o}\"{o}s, Mika and Hollender, Alexandros and Jain, Siddhartha and Maystre, Gilbert and Pires, William and Robere, Robert and Tao, Ran},
  title =	{{Further Collapses in TFNP}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{33:1--33:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.33},
  URN =		{urn:nbn:de:0030-drops-165954},
  doi =		{10.4230/LIPIcs.CCC.2022.33},
  annote =	{Keywords: TFNP, PPAD, PLS, EOPL}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2019.65

The Hairy Ball Problem is PPAD-Complete

Authors: Paul W. Goldberg and Alexandros Hollender

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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.

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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)

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@InProceedings{goldberg_et_al:LIPIcs.ICALP.2019.65,
  author =	{Goldberg, Paul W. and Hollender, Alexandros},
  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 =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.65},
  URN =		{urn:nbn:de:0030-drops-106416},
  doi =		{10.4230/LIPIcs.ICALP.2019.65},
  annote =	{Keywords: Computational Complexity, TFNP, PPAD, End-of-Line}
}

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Further Collapses in TFNP

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The Hairy Ball Problem is PPAD-Complete

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