License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.11
URN: urn:nbn:de:0030-drops-178610
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17861/
Bandyapadhyay, Sayan ;
Lochet, William ;
Saurabh, Saket ;
Xue, Jie
Minimum-Membership Geometric Set Cover, Revisited
Abstract
We revisit a natural variant of the geometric set cover problem, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set S of points and a set ℛ of geometric objects, and the goal is to find a subset ℛ^* ⊆ ℛ to cover all points in S such that the membership of S with respect to ℛ^*, denoted by memb(S,ℛ^*), is minimized, where memb(S,ℛ^*) = max_{p ∈ S} |{R ∈ ℛ^*: p ∈ R}|. We give the first polynomial-time approximation algorithms for MMGSC in ℝ². Specifically, we achieve the following two main results.
- We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time n^{O(opt)} where opt is the optimum of the problem (i.e., the minimum membership).
- We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of o(log n) in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find ℛ^* ⊆ ℛ to cover S such that the ply of ℛ^* is minimized, where the ply is defined as the maximum number of objects in ℛ^* which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in O(n^{12}) time. We give a significantly simpler constant-approximation algorithm with near-linear running time.
BibTeX - Entry
@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.11,
author = {Bandyapadhyay, Sayan and Lochet, William and Saurabh, Saket and Xue, Jie},
title = {{Minimum-Membership Geometric Set Cover, Revisited}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17861},
URN = {urn:nbn:de:0030-drops-178610},
doi = {10.4230/LIPIcs.SoCG.2023.11},
annote = {Keywords: geometric set cover, geometric optimization, approximation algorithms}
}
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Keywords: |
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geometric set cover, geometric optimization, approximation algorithms |
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Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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09.06.2023 |
