eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
70:1
70:16
10.4230/LIPIcs.MFCS.2024.70
article
On Line-Separable Weighted Unit-Disk Coverage and Related Problems
Liu, Gang
1
Wang, Haitao
1
Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.70/LIPIcs.MFCS.2024.70.pdf
Line-separable
unit disks
halfplanes
geometric coverage
geometric hitting set