Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem

TL;DR

This paper explores the computational complexity of approximate solutions to the consensus halving problem, linking it to the Borsuk-Ulam theorem and introducing new complexity classes to better understand their relationships.

Contribution

It establishes polynomial time equivalence between approximate consensus halving solutions and Borsuk-Ulam search problems, and introduces the complexity class BBU for alternative formulations.

Findings

Polynomial time equivalence between consensus halving approximation and Borsuk-Ulam search.

Introduction of the new complexity class BBU for Borsuk-Ulam related problems.

Structural analysis of BBU, BU, and FIXP complexity classes.

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 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 $\varepsilon$-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.