On Line-Separable Weighted Unit-Disk Coverage and Related Problems

TL;DR

This paper presents improved algorithms for line-separable weighted disk coverage problems, achieving faster solutions for covering points with disks or halfplanes, and extends to related hitting set problems.

Contribution

The authors develop an $O(n^{3/2} ext{log}^2 n)$ time algorithm for line-separable unit-disk coverage, improving upon previous methods and enabling faster solutions for related geometric covering problems.

Findings

Achieved $O(n^{3/2} ext{log}^2 n)$ time complexity for the line-separable unit-disk coverage problem.

Improved the halfplane coverage problem solution to $O(n^{7/2} ext{log}^2 n)$ time.

Extended the approach to solve hitting set problems with similar efficiency.

Abstract

Given a set $P$ of $n$ points and a set $S$ of $n$ weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of $P$. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of $P$ by a line $\ell$. We present an $O(n^{3/2}\log^2 n)$ time algorithm for the problem. This improves the previously best work of $O(n^2\log n)$ time. Our result leads to an algorithm of $O(n^{{7}/{2}}\log^2 n)$ time for the halfplane coverage problem (i.e., using $n$ weighted halfplanes to cover $n$ points), an improvement over the previous $O(n^4\log n)$ time solution. If all halfplanes are lower ones, our algorithm runs in $O(n^{{3}/{2}}\log^2 n)$ time, while the previous best algorithm takes $O(n^2\log n)$ time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.