Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem

TY - GEN

T1 - Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem

AU - Batziou, Eleni

AU - Hansen, Kristoffer Arnsfelt

AU - Høgh, Kasper

N1 - Conference code: 48

PY - 2021/7/1

Y1 - 2021/7/1

N2 - In the consensus halving problem we are given n agents with valuations over the interval [0, 1]. The goal is to divide the interval into at most n + 1 pieces (by placing at most n cuts), which may be combined to give a partition of [0, 1] into two sets valued equally by all agents. The existence of a solution may be established by the Borsuk-Ulam theorem. We consider the task of computing an approximation of an exact solution of the consensus halving problem, where the valuations are given by distribution functions computed by algebraic circuits. Here approximation refers to computing a point that is ∈-close to an exact solution, also called strong approximation. We show that this task is polynomial time equivalent to computing an approximation to an exact solution of the Borsuk-Ulam search problem defined by a continuous function that is computed by an algebraic circuit. The Borsuk-Ulam search problem is the defining problem of the complexity class BU. We introduce a new complexity class BBU to also capture an alternative formulation of the Borsuk-Ulam theorem from a computational point of view. We investigate their relationship and prove several structural results for these classes as well as for the complexity class FIXP.

AB - In the consensus halving problem we are given n agents with valuations over the interval [0, 1]. The goal is to divide the interval into at most n + 1 pieces (by placing at most n cuts), which may be combined to give a partition of [0, 1] into two sets valued equally by all agents. The existence of a solution may be established by the Borsuk-Ulam theorem. We consider the task of computing an approximation of an exact solution of the consensus halving problem, where the valuations are given by distribution functions computed by algebraic circuits. Here approximation refers to computing a point that is ∈-close to an exact solution, also called strong approximation. We show that this task is polynomial time equivalent to computing an approximation to an exact solution of the Borsuk-Ulam search problem defined by a continuous function that is computed by an algebraic circuit. The Borsuk-Ulam search problem is the defining problem of the complexity class BU. We introduce a new complexity class BBU to also capture an alternative formulation of the Borsuk-Ulam theorem from a computational point of view. We investigate their relationship and prove several structural results for these classes as well as for the complexity class FIXP.

KW - Borsuk-Ulam

KW - Computational Complexity

KW - Consensus halving

UR - http://www.scopus.com/inward/record.url?scp=85112690498&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2021.24

DO - 10.4230/LIPIcs.ICALP.2021.24

M3 - Article in proceedings

SN - 978-3-95977-195-5

VL - 198

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 24:1-24:20

BT - 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

A2 - Bansal, Nikhil

A2 - Merelli, Emanuela

A2 - Worrell, James

PB - Dagstuhl Publishing

T2 - 48th International Colloquium on Automata, Languages<br/>and Programming

Y2 - 12 July 2021 through 16 July 2021

ER -