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.2019.138
URN: urn:nbn:de:0030-drops-107141
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10714/
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Deligkas, Argyrios ; Fearnley, John ; Melissourgos, Themistoklis ; Spirakis, Paul G.

Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem

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Abstract

We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete [Filos-Ratsikas and Goldberg, 2018; Filos-Ratsikas and Goldberg, 2018], we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP-hard, and deciding whether there exists a solution with fewer than n cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial.
Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP subseteq BU subseteq TFETR and that LinearBU = PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.

BibTeX - Entry

@InProceedings{deligkas_et_al:LIPIcs:2019:10714,
  author =	{Argyrios Deligkas and John Fearnley and Themistoklis Melissourgos and Paul G. Spirakis},
  title =	{{Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{138:1--138:14},
  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/2019/10714},
  URN =		{urn:nbn:de:0030-drops-107141},
  doi =		{10.4230/LIPIcs.ICALP.2019.138},
  annote =	{Keywords: PPA, FIXP, ETR, consensus halving, circuit, reduction, complexity class}
}

Keywords: PPA, FIXP, ETR, consensus halving, circuit, reduction, complexity class
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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