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

Authors Gang Liu, Haitao Wang

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LIPIcs.MFCS.2024.68.pdf
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Author Details

Gang Liu
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA

Funding

This research was supported in part by NSF under Grant CCF-2300356.

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Gang Liu and Haitao Wang. Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.68

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. This paper considers a line-constrained version in which all disks have their centers on a line. We present an O(mlog²n+(n+m)log(n+m)) time algorithm for the problem. This improves the previous result of O(m²log 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 also 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 𝓁 and the boundary of every two disks intersect at most once on the side of 𝓁 containing P.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • hitting set
  • line-constrained
  • line-separable
  • unit disks
  • half-planes
  • coverage

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