In the $\varepsilon$-Consensus-Halving problem, a fundamental problem in fair division, there are $n$ agents with valuations over the interval $[0,1]$, and the goal is to divide the interval into pieces and assign a label "$+$" or "$-$" to each piece, such that every agent values the total amount of "$+$" and the total amount of "$-$" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving $\varepsilon$-Consensus-Halving for any $\varepsilon$, as well as a polynomial-time algorithm for $\varepsilon=1/2$. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-$1/k$-Division problem [Simmons and Su, 2003]. In particular, we prove that $\varepsilon$-Consensus-$1/3$-Division is PPAD-hard.

翻译：在$\varepsilon$-共识一半问题中，一个公平分配中的基本问题，存在$n个$对区间$[0,1]$具有估值的智能体，目标是将该区间划分为若干片段，并为每个片段分配标签“$+$”或“$-$”，使得每个智能体对“$+$”总量和“$-$$”总量的估值几乎相等。该问题最近由Filos-Ratsikas和Goldberg [2019]证明是计算类PPA中第一个“自然”完全问题，解答了长达十年的未解之谜。在本文中，我们探讨了若限制估值函数类别，问题在多大程度上变得容易解决。为此，我们做出以下贡献。首先，我们加强了[Filos-Ratsikas和Goldberg, 2019]的PPA硬度结果，将其扩展至智能体仅具有两个区块的分段均匀估值的情形。我们通过一种新的归约方法获得此结果，该方法在概念上实际上比[Filos-Ratsikas和Goldberg, 2019]中对应的归约简单得多。接着，我们考虑单区块（均匀）估值的情况，并提出了一个参数化多项式时间算法，用于求解任意$\varepsilon$下的$\varepsilon$-共识一半问题，以及一个针对$\varepsilon=1/2$的多项式时间算法。最后，我们新技巧的一个重要应用是得到共识一半问题推广形式——共识-$1/k$-分割问题[Simmons和Su, 2003]的首个硬度结果。特别地，我们证明了$\varepsilon$-共识-$1/3$-分割问题是PPAD-hard的。