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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.32

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

Funding

  • Mitchell, Joseph S. B.: Partially supported by the National Science Foundation (CCF-1540890), the US-Israel Binational Science Foundation (CCF-1540890), and DARPA (Lagrange).
  • Mustafa, Nabil H.: The work of Nabil H. Mustafa in this paper has been supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).

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