The Complexity of Hex and the Jordan Curve Theorem
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The complexity of hex.pdf
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1.15 MB
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Author(s)
Adler, Aviv
Daskalakis, Konstantinos
Demaine, Erik D
Date Issued
July 2016
Journal
43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Publisher
Schloss Dagstuhl, Leibniz-Zentrum für Informatik GmbH
Citation
Adler, Aviv, Constantinos Daskalakis, and Erik Demaine. "The Complexity of Hex and the Jordan Curve Theorem" In 43rd International Colloquium on Automata, Languages, and Programming: ICALP 2016, Rome, Italy, July 12-15, 2016. Article no. 24; pp. 24:1-24:14.
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution 4.0 International License
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.ICALP.2016.24
