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

## TL;DR

This paper investigates the computational complexity of finding exact solutions to the consensus halving problem, revealing it to be significantly harder than the approximate version, and introduces a new complexity class related to the Borsuk-Ulam theorem.

## Contribution

It establishes the complexity classifications of the exact consensus halving problem and introduces the BU complexity class related to Borsuk-Ulam problems.

## Key findings

- Exact consensus halving is FIXP-hard.
- Deciding fewer than n cuts is ETR-complete.
- Introduces the BU complexity class and its relation to FIXP, PPA, and TFETR.

## 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 $n$ 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.