Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint

Authors Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Joseph S. B. Mitchell, Nabil H. Mustafa

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Author Details

Chien-Chung Huang
  • DI ENS, École Normale supérieure, Université PSL, Paris, France
  • CNRS, Paris, France
Mathieu Mari
  • DI ENS, École Normale supérieure, Université PSL, Paris, France
Claire Mathieu
  • CNRS, Paris, France
Joseph S. B. Mitchell
  • Stony Brook University, Stony Brook, NY 11794, USA
Nabil H. Mustafa
  • Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France

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Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Joseph S. B. Mitchell, and Nabil H. Mustafa. Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Given a set D of n unit disks in the plane and an integer k <= n, the maximum area connected subset problem asks for a set D' subseteq D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint. We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 1/2-approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of epsilon k disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • approximation algorithm
  • submodular function optimisation
  • unit disk graph
  • connectivity constraint


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  1. Sanjeev Arora. Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and Other Geometric Problems. J. ACM, 45(5):753-782, September 1998. URL:
  2. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a Monotone Submodular Function Subject to a Matroid Constraint. SIAM J. Comput., 40(6):1740-1766, 2011. Google Scholar
  3. Erin W. Chambers, Sándor P. Fekete, Hella-Franziska Hoffmann, Dimitri Marinakis, Joseph S.B. Mitchell, Venkatesh Srinivasan, Ulrike Stege, and Sue Whitesides. Connecting a set of circles with minimum sum of radii. Computational Geometry, 68(1-3):62-76, January 1991. special issue in memory of Ferran Hurtado. Google Scholar
  4. Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Approximation Schemes for Geometric Coverage Problems. In ESA, volume 112 of LIPIcs, pages 17:1-17:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  5. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes. SIAM J. Comput., 43(6):1831-1879, 2014. Google Scholar
  6. Yuval Filmus and Justin Ward. Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search. SIAM J. Comput., 43(2):514-542, 2014. URL:
  7. Stefan Funke, Alex Kesselman, Fabian Kuhn, Zvi Lotker, and Michael Segal. Improved Approximation Algorithms for Connected Sensor Cover. Wirel. Netw., 13(2):153-164, April 2007. URL:
  8. L. A. Wolsey G.L. Nemhauser and M.L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Mathematical Programming, 14(1):265-294, 1978. Google Scholar
  9. Himanshu Gupta, Zongheng Zhou, Samir R. Das, and Quinyi Gu. Connected Sensor Cover: Self-organization of Sensor Networks for Efficient Query Execution. IEEE/ACM Trans. Netw., 14(1):55-67, February 2006. URL:
  10. Dorit S. Hochbaum and Anu Pathria. Node-Optimal Connected k-Subgraphs, 1994. Google Scholar
  11. Koushik Kar and Suman Banerjee. Node Placement for Connected Coverage in Sensor Networks, 2003. Google Scholar
  12. Samir Khuller, Manish Purohit, and Kanthi K. Sarpatwar. Analyzing the Optimal Neighborhood: Algorithms for Budgeted and Partial Connected Dominating Set Problems. In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1702-1713, Philadelphia, PA, USA, 2014. Society for Industrial and Applied Mathematics. URL:
  13. Ariel Kulik, Hadas Shachnai, and Tami Tamir. Maximizing submodular set functions subject to multiple linear constraints. In SODA, pages 545-554. SIAM, 2009. Google Scholar
  14. Tung-Wei Kuo, Kate Ching-Ju Lin, and Ming-Jer Tsai. Maximizing Submodular Set Function with Connectivity Constraint: Theory and Application to Networks. IEEE/ACM Trans. Netw., 23(2):533-546, April 2015. URL:
  15. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular Maximization over Multiple Matroids via Generalized Exchange Properties. Math. Oper. Res., 35(4):795-806, 2010. Google Scholar
  16. M. V. Marathe, H. Breu, H. B. Hunt III, S. S. Ravi, and D. J. Rosenkrantz. Simple heuristics for unit disk graphs. NETWORKS, 1995. Google Scholar
  17. Tomomi Matsui. Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs. In Jin Akiyama, Mikio Kano, and Masatsugu Urabe, editors, Discrete and Computational Geometry, pages 194-200, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. Google Scholar
  18. Nabil H. Mustafa and Saurabh Ray. Improved Results on Geometric Hitting Set Problems. Discrete & Computational Geometry, 44(4):883-895, 2010. Google Scholar
  19. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, New York, NY, USA, 2007. Google Scholar
  20. Yuval Rabani and Gabriel Scalosub. Bicriteria approximation tradeoff for the node-cost budget problem. ACM Trans. Algorithms, 5(2):19:1-19:14, 2009. Google Scholar
  21. Fabio Vandin, Eli Upfal, and Benjamin J. Raphael. Algorithms for Detecting Significantly Mutated Pathways in Cancer. Journal of Computational Biology, 18(3):507-522, 2011. Google Scholar
  22. Vijay V. Vazirani. Euclidean TSP, pages 84-89. Springer Berlin Heidelberg, Berlin, Heidelberg, 2003. URL:
  23. L. Wolsey. Maximising real-valued submodular functions: primal and dual heuristics for location problems. Mathematics of Operations Research, 7(3):410-425, 1982. Google Scholar
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