Exact algorithms for optimal discretization

2026-08-25

AI Summary

本文针对最优离散化问题，提出了一种运行时间为O(1.9602^n)的算法，并对点分离问题提出了O(1.8906^n)的算法，通过结构观察改进了基本的2^n时间复杂度。

Abstract

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.

Problem

• optimal discretization
• point separation
• parameterized algorithmics
• supervised machine learning

Innovation

• Optimal Discretization
• Point Separation
• Algorithmic Improvement