共识平分：它会变得更容易吗？
Consensus-Halving: Does It Ever Get Easier?
AI总结：
本文研究限制估值类别能否降低 ε-Consensus-Halving 难度，证明两块分段一致估值仍 PPA 困难，给出单块估值的参数化及 ε=1/2 多项式算法，并证明 ε-Consensus-1/3-Division 为 PPAD 困难。
摘要：
在 ε-Consensus-Halving（ε-共识平分）问题这一公平分割中的基础性问题中，有 n 个对区间 [0,1] 具有估值的智能体，目标是把该区间分割成若干块，并为每一块分配“+”或“-”标签，使每个智能体对“+”总量和“-”总量的估值几乎相等。Filos-Ratsikas 和 Goldberg [2019] 最近证明，该问题是计算类 PPA 的第一个“自然的”完全问题，回答了一个存在十年之久的开放问题。本文考察如果限制估值函数的类别，该问题在多大程度上会变得容易求解。为此，我们给出以下贡献。首先，我们把 [Filos-Ratsikas and Goldberg, 2019] 的 PPA 困难性结果加强到智能体只具有两块分段一致估值的情形。我们通过一种新的归约获得该结果，事实上该归约在概念上比 [Filos-Ratsikas and Goldberg, 2019] 中相应的归约简单得多。随后，我们考虑单块（一致）估值情形，并针对任意 ε 给出求解 ε-Consensus-Halving 的参数化多项式时间算法，以及针对 ε=1/2 的多项式时间算法。最后，我们新技术的一项重要应用是得到 Consensus-Halving 的推广问题 Consensus-1/k-Division（共识 1/k 分割）问题 [Simmons and Su, 2003] 的首个困难性结果。具体而言，我们证明 ε-Consensus-1/3-Division 是 PPAD 困难的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
