Minimum-Membership Geometric Set Cover, Revisited

Authors Sayan Bandyapadhyay, William Lochet, Saket Saurabh, Jie Xue



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Author Details

Sayan Bandyapadhyay
  • Portland State University, OR, USA
William Lochet
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
Jie Xue
  • New York University Shanghai, China

Acknowledgements

The authors would like to thank Qizheng He, Daniel Lokshtanov, Rahul Saladi, Subhash Suri, and Haitao Wang for helpful discussions about the problems, and thank the anonymous reviewers for their detailed comments, which help significantly improve the writing of the paper.

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Sayan Bandyapadhyay, William Lochet, Saket Saurabh, and Jie Xue. Minimum-Membership Geometric Set Cover, Revisited. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.11

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • geometric set cover
  • geometric optimization
  • approximation algorithms

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References

  1. Pankaj Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic geometric set cover and hitting set. ACM Transactions on Algorithms (TALG), 18(4):1-37, 2022. Google Scholar
  2. Pankaj K Agarwal and Jiangwei Pan. Near-linear algorithms for geometric hitting sets and set covers. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 271-279, 2014. Google Scholar
  3. Therese Biedl, Ahmad Biniaz, and Anna Lubiw. Minimum ply covering of points with disks and squares. Comput. Geom., 94:101712, 2021. URL: https://doi.org/10.1016/j.comgeo.2020.101712.
  4. Timothy M Chan and Qizheng He. Faster approximation algorithms for geometric set cover. In 36th International Symposium on Computational Geometry (SoCG 2020), volume 164, page 27. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2020. Google Scholar
  5. Stephane Durocher, J Mark Keil, and Debajyoti Mondal. Minimum ply covering of points with unit squares. In WALCOM: Algorithms and Computation: 17th International Conference and Workshops, WALCOM 2023, Hsinchu, Taiwan, March 22-24, 2023, Proceedings, pages 23-35. Springer, 2023. Google Scholar
  6. Thomas Erlebach and Erik Jan van Leeuwen. Approximating geometric coverage problems. In Shang-Hua Teng, editor, Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 1267-1276. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347220.
  7. Fabian Kuhn, Pascal von Rickenbach, Roger Wattenhofer, Emo Welzl, and Aaron Zollinger. Interference in cellular networks: The minimum membership set cover problem. In Lusheng Wang, editor, Computing and Combinatorics, 11th Annual International Conference, COCOON 2005, Kunming, China, August 16-29, 2005, Proceedings, volume 3595 of Lecture Notes in Computer Science, pages 188-198. Springer, 2005. URL: https://doi.org/10.1007/11533719_21.
  8. Joseph SB Mitchell and Supantha Pandit. Minimum membership covering and hitting. Theoretical Computer Science, 876:1-11, 2021. Google Scholar
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