We consider the ε-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to ε). This problem was recently shown to be PPA-complete, for n agents and n cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem.

For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.

我们考虑ε - Consensus-Halving问题，其中一组异构代理旨在将连续资源分成两个（不一定是连续的）部分，所有这些部分同时认为具有大致相同的值（直到ε）。这个问题最近被证明是 PPA 完全的，对于n 个代理和n 个割集，即使对于非常简单的估值函数也是如此。为了了解问题复杂性的根源，我们考虑了只有恒定数量的代理的设置，并且我们同时考虑了问题的计算复杂度和查询复杂度。

对于具有单调估值函数的代理，我们展示了一个二分法：对于两个代理，问题是多项式时间可解决的，而对于三个或更多代理，它变成 PPA 完全的。类似地，我们表明，对于两个单调代理，可以通过多项式多查询来解决问题，而对于三个或更多代理，我们提供指数查询复杂度下限。这些结果是通过与单调 Borsuk-Ulam问题的有趣联系来实现的，这可能是独立感兴趣的。对于具有一般估值的代理，我们证明问题是 PPA 完全的，并且承认指数查询复杂度下限，即使对于两个代理也是如此。