The Complexity of Hex and the Jordan Curve Theorem
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 theore...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Schloss Dagstuhl, Leibniz-Zentrum für Informatik GmbH
2017
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| Online Access: | http://hdl.handle.net/1721.1/110842 https://orcid.org/0000-0003-3698-7639 https://orcid.org/0000-0002-5451-0490 https://orcid.org/0000-0003-3803-5703 |
| Summary: | 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. |
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