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

19^th June 2019, 13:00
Argyrios Deligkas
University of Liverpool

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 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 ? BU ? 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.

Joint work with John Fearnley, Themistoklis Melissourgos and Paul Spirakis.
To appear in ICALP '19.