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

Authors Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos , Paul G. Spirakis



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Argyrios Deligkas
  • Department of Computer Science, University of Liverpool, Liverpool, UK
  • Leverhulme Research Centre for Functional Materials Design, Liverpool, UK
John Fearnley
  • Department of Computer Science, University of Liverpool, Liverpool, UK
Themistoklis Melissourgos
  • Department of Computer Science, University of Liverpool, Liverpool, UK
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, Liverpool, UK
  • Computer Engineering and Informatics Department, University of Patras, Patras, Greece

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Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis. Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 138:1-138:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.138

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • PPA
  • FIXP
  • ETR
  • consensus halving
  • circuit
  • reduction
  • complexity class

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