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 [29], [30], 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. Along the way, we define a new complexity class, called BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that $\mathtt{FIXP}\subseteq \mathtt{BU}\subseteq \mathtt{TFETR}$ and that $\mathtt{LinearBU}=\mathtt{PPA}$, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.

我们研究为共识减半问题找到精确解决方案的问题。尽管最近的工作表明此问题的近似版本是PPA -complete [29]，[30]，但我们表明确切的版本要困难得多。具体地，找到一个解决方案Ñ剂和Ñ切口被FIXP -hard，并决定是否存在少于溶液Ñ切口是ETR -complete。在此过程中，我们定义了一个称为BU的新的复杂性类，该类捕获了可以简化为完全解决Borsuk-Ulam问题实例的所有问题。我们证明$\mathtt{FIXP}\subseteq \mathtt{卜}\subseteq \mathtt{TFETR}$ 然后 $\mathtt{线性BU}=\mathtt{PPA}$，其中LinearBU是BU的子类，其中BORsuk -Ulam实例由线性算术电路指定。