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A PTAS for the Weighted Unit Disk Cover Problem

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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Abstract

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

Research supported in part by the National Basic Research Program of China Grant 2015CB358700, 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61202009, 61033001, 61361136003.

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Authors and Affiliations

  1. IIIS, Tsinghua University, Beijing, China

    Jian Li & Yifei Jin

Authors
  1. Jian Li
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  2. Yifei Jin
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Correspondence to Yifei Jin .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Li, J., Jin, Y. (2015). A PTAS for the Weighted Unit Disk Cover Problem. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_73

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_73

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47671-0

  • Online ISBN: 978-3-662-47672-7

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