AI 中文总结
本文针对最优离散化问题提出运行时间O(1.9602ⁿ)的精确算法，针对轴平行点分离问题提出O(1.8906ⁿ)的算法，首次改进了两问题的平凡2ⁿ界限。
AI 中文摘要
最优离散化问题是指，给定平面上两个不相交的点集R和B，求一个由最少数量的水平线和竖直线组成的集合，使得这些线划分出的每个单元格都不包含来自两个集合的点，即所有单元格仅包含R或仅包含B中的点。该问题是监督机器学习中的一种预处理步骤，在参数化算法学领域已受到大量关注。针对Bonnet、Giannopoulos和Lampis[IPEC 2017]以及Froese[博士论文，2018]提出的问题，Kratsch、Masařík、Muzi、Pilipczuk和Sorge[SODA 2021]证明最优离散化问题存在一个固定参数算法，其运行时间为2^{O(k² log k)}·n^{O(1)}，其中k是解的大小，n = |R| + |B|。在本文中，我们给出了一个最优离散化算法，其运行时间为O(1.9602ⁿ)。我们还研究了相关的点分离问题，该问题要求用轴平行线分离所有输入点。对于这个问题，我们得到了一个运行时间为O(1.8906ⁿ)的算法。我们的算法保证基于双色点集和单色点集的结构观察，即使允许点共享坐标，这些保证仍然成立。据我们所知，这是两个问题首次对平凡的2ⁿ界限做出改进。
英文摘要
The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8906^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets, and hold even if points are allowed to share coordinates. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.
CommentsTo appear in IPEC 2026
