$α$ -Ham-Sandwich问题的计算复杂性
Computational Complexity of the $α$-Ham-Sandwich Problem
摘要：
经典的Ham-Sandwich定理指出，对于$\mathbb{R}^d$中的任何$d$可测集，存在一个同时平分它们的超平面。Bárány，Hubard和Jerónimo [DCG 2008]的扩展指出，如果集合是凸的并且是\emph{well-separated}，则对于任何给定的$α_1, \dots, α_d \in [0, 1]$，存在唯一的定向超平面，该超平面从每个集合中截断相应的分数$α_1, \dots, α_d$。Steiger和Zhao [DCG 2010]证明了这个定理的离散模拟，我们称之为\emph{$α$-Ham-Sandwich theorem}。他们给出了一个在时间上找到超平面的算法$O(n (\log n)^{d-3})$，其中$n$是输入点的总数。这个搜索问题在高维中的计算复杂性是开放的，与Ham-Sandwich问题的复杂性完全不同，Ham-Sandwich问题现在已知是PPA完全的（Filos-Ratsikas和Goldberg [STOC 2019]）。最近，Fearley，Gordon，Mehta和Savani [ICALP 2019]引入了CLS（连续本地搜索）的一个新子类，称为\emph{Unique End-of-Potential Line}（UEOPL）。这个类捕获CLS中具有独特解决方案的问题。证明了对于$α$-Ham-Sandwich定理，寻找可分超平面的搜索问题存在于UEOPL中。这给出了第一个非平凡的包容性的问题，在一个复杂的类，并将其放在公司的经典搜索问题，如寻找不动点的收缩映射，唯一的汇定向问题和$P$ -矩阵线性互补问题。
英文摘要：
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $\mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and \emph{well-separated}, then for any given $α_1, \dots, α_d \in [0, 1]$, there is a unique oriented hyperplane that cuts off a respective fraction $α_1, \dots, α_d$ from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the \emph{$α$-Ham-Sandwich theorem}. They gave an algorithm to find the hyperplane in time $O(n (\log n)^{d-3})$, where $n$ is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called \emph{Unique End-of-Potential Line} (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the $α$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the $P$-matrix linear complementarity problem.
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