Hardness Results for Consensus-Halving

Authors Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, Jie Zhang

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Aris Filos-Ratsikas
  • École Polytechnique Fédérale de Lausanne, Switzerland
Søren Kristoffer Stiil Frederiksen
  • Aarhus University, Denmark
Paul W. Goldberg
  • University of Oxford, United Kingdom
Jie Zhang
  • University of Southampton, United Kingdom

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Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang. Hardness Results for Consensus-Halving. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • PPAD
  • PPA
  • consensus halving
  • generalized-circuit
  • reduction


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