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

Argyrios Deligkas; John Fearnley; Themistoklis Melissourgos; Paul G. Spirakis

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arXiv · 1903.03101

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

Argyrios Deligkas·John Fearnley·Themistoklis Melissourgos·Paul G. Spirakis

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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.

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Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, Paul G. Spirakis. 2019-03-07. Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem. https://arxiv.org/abs/1903.03101

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