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

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LIPIcs.MFCS.2024.70.pdf
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## Cite As

Gang Liu and Haitao Wang. On Line-Separable Weighted Unit-Disk Coverage and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.70

## 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 𝓁. We present an O(n^{3/2}log² n) time algorithm for the problem. This improves the previously best work of O(n²log n) time. Our result leads to an algorithm of O(n^{7/2}log² n) time for the halfplane coverage problem (i.e., using n weighted halfplanes to cover n points), an improvement over the previous O(n⁴log n) time solution. If all halfplanes are lower ones, our algorithm runs in O(n^{3/2}log² n) time, while the previous best algorithm takes O(n²log n) time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Line-separable
• unit disks
• halfplanes
• geometric coverage
• geometric hitting set

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