Batziou, Eleni ;
Hansen, Kristoffer Arnsfelt ;
Høgh, Kasper
Strong Approximate Consensus Halving and the BorsukUlam Theorem
Abstract
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 BorsukUlam 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 εclose 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 BorsukUlam search problem defined by a continuous function that is computed by an algebraic circuit.
The BorsukUlam 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 BorsukUlam 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.
BibTeX  Entry
@InProceedings{batziou_et_al:LIPIcs.ICALP.2021.24,
author = {Batziou, Eleni and Hansen, Kristoffer Arnsfelt and H{\o}gh, Kasper},
title = {{Strong Approximate Consensus Halving and the BorsukUlam Theorem}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {24:124:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771955},
ISSN = {18688969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14093},
URN = {urn:nbn:de:0030drops140939},
doi = {10.4230/LIPIcs.ICALP.2021.24},
annote = {Keywords: Consensus halving, Computational Complexity, BorsukUlam}
}
02.07.2021
Keywords: 

Consensus halving, Computational Complexity, BorsukUlam 
Seminar: 

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

Issue date: 

2021 
Date of publication: 

02.07.2021 