License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ICALP.2016.24
URN: urn:nbn:de:0030-drops-63032
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Adler, Aviv ; Daskalakis, Constantinos ; Demaine, Erik D.

The Complexity of Hex and the Jordan Curve Theorem

LIPIcs-ICALP-2016-24.pdf (1 MB)


The Jordan curve theorem and Brouwer's fixed-point theorem are fundamental problems in topology. We study their computational relationship, showing that a stylized computational version of Jordan’s theorem is PPAD-complete, and therefore in a sense computationally equivalent to Brouwer’s theorem. As a corollary, our computational result implies that these two theorems directly imply each other mathematically, complementing Maehara's proof that Brouwer implies Jordan [Maehara, 1984]. We then turn to the combinatorial game of Hex which is related to Jordan's theorem, and where the existence of a winner can be used to show Brouwer's theorem [Gale,1979]. We establish that determining who won an (implicitly encoded) play of Hex is PSPACE-complete by adapting a reduction (due to Goldberg [Goldberg,2015]) from Quantified Boolean Formula (QBF). As this problem is analogous to evaluating the output of a canonical path-following algorithm for finding a Brouwer fixed point - and which is known to be PSPACE-complete [Goldberg/Papadimitriou/Savani, 2013] - we thereby establish a connection between Brouwer, Jordan and Hex higher in the complexity hierarchy.

BibTeX - Entry

  author =	{Aviv Adler and Constantinos Daskalakis and Erik D. Demaine},
  title =	{{The Complexity of Hex and the Jordan Curve Theorem}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-63032},
  doi =		{10.4230/LIPIcs.ICALP.2016.24},
  annote =	{Keywords: Jordan, Brouwer, Hex, PPAD, PSPACE}

Keywords: Jordan, Brouwer, Hex, PPAD, PSPACE
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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