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

Deligkas, Argyrios, Fearnley, John, Melissourgos, Themistoklis ORCID: 0000-0002-9867-6257 and 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, we show that the exact version is much harder. Specifically, finding a solution with $n$ 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.

Item Type:	Article
Uncontrolled Keywords:	cs.CC, cs.CC
Depositing User:	Symplectic Admin
Date Deposited:	11 Mar 2019 08:38
Last Modified:	19 Jan 2023 00:57
Open Access URL:	https://arxiv.org/abs/1903.03101
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3033986

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• Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem. (deposited 11 Mar 2019 08:38) [Currently Displayed]
  • Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem. (deposited 30 Jan 2020 14:08)