2026-07-23T17:33:45Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10714 2024-03-06T10:46:32Z ddc:004 open_access

Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem Deligkas, Argyrios Fearnley, John Melissourgos, Themistoklis Spirakis, Paul G. PPA FIXP ETR consensus halving circuit reduction complexity class 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Argyrios Deligkas and John Fearnley and Themistoklis Melissourgos and Paul G. Spirakis 2019 Is Part Of LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2019.138 urn:nbn:de:0030-drops-107141 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.138 eng https://creativecommons.org/licenses/by/3.0/legalcode