Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems

TL;DR

This paper introduces a faster algorithm for a line-constrained disk hitting set problem, improving computational efficiency for a geometric problem with applications in computational geometry.

Contribution

It presents an $O(m ext{log}^2 n + (n+m) ext{log}(n+m))$ time algorithm for a line-constrained disk hitting set problem, extending to a more general line-separable case.

Findings

Improved algorithm reduces time complexity for the problem.

Solves a more general line-separable problem with a single intersection property.

Enhances efficiency over previous methods for weighted disk hitting set problems.

Abstract

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 $P$.