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: https://drops.dagstuhl.de/opus/volltexte/2023/17861/
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Bandyapadhyay, Sayan ; Lochet, William ; Saurabh, Saket ; Xue, Jie

Minimum-Membership Geometric Set Cover, Revisited

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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}
}

Keywords: geometric set cover, geometric optimization, approximation algorithms
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023


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