线约束圆盘的无权几何命中集及相关问题
Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems
AI总结：
本文针对圆心共线的圆盘命中集问题提出O(m log² n+(n+m)log(n+m))时间算法，改进加权情形的最好结果，并推广到具有单一相交性质的线可分问题。
摘要：
给定平面上由n个点组成的集合P和由m个圆盘组成的集合S，圆盘命中集问题要求找到P的一个最小子集，使得S中的每个圆盘至少包含该子集中的一个点。该问题是NP难的。本文考虑一种线约束版本，其中所有圆盘的圆心都位于一条直线上。我们为该问题提出了一个O(m log² n + (n+m) log(n+m))时间的算法。这改进了此前该问题加权情形下O(m² log m + (n+m) log(n+m))时间的最好结果，在加权情形中P的每个点都有一个权重，目标是使命中集的总权重最小化。我们的算法实际上解决了一个更一般的具有单一相交性质的线可分问题：P中的点与圆盘圆心被一条直线ℓ分隔，并且任意两个圆盘的边界在ℓ包含P的一侧至多相交一次。
英文摘要：
Given a set $P$ of $n$ points and a set $S$ of $m$ disks in the plane, the disk hitting set problem asks for a smallest subset of $P$ such that every disk of $S$ contains at least one point in the subset. The problem is NP-hard. In this paper, we consider a line-constrained version in which all disks have their centers on a line. We present an $O(m\log^2n+(n+m)\log(n+m))$ time algorithm for the problem. This improves the previously best result of $O(m^2\log m+(n+m)\log(n+m))$ time for the weighted case of the problem where every point of $P$ has a weight and the objective is to minimize the total weight of the hitting set. Our algorithm actually solves a more general line-separable problem with a single intersection property: The points of $P$ and the disk centers are separated by a line $\ell$ and the boundary of every two disks intersect at most once on the side of $\ell$ containing $P$.
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。
