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PTAS for Weighted Set Cover on Unit Squares

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Book cover Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

Abstract

We study the planar version of Minimum-Weight Set Cover, where one has to cover a given set of points with a minimum-weight subset of a given set of planar objects. For the unit-weight case, one PTAS (on disks) is known. For arbitrary weights however, the problem appears much harder, and in particular no PTASs are known. We present the first PTAS for Weighted Geometric Set Cover on planar objects, namely on axis-parallel unit squares. By extending the algorithm, we also obtain a PTAS for Minimum-Weight Dominating Set on intersection graphs of unit squares and Geometric Budgeted Maximum Coverage on unit squares. The running time of the developed algorithms is optimal under the exponential time hypothesis. We also show inapproximability results for Geometric Set Cover on various object shapes that are more general than unit squares.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Thomas Erlebach

  2. Department of Informatics, University of Bergen, P.O. Box 7803, N-5020, Bergen, Norway

    Erik Jan van Leeuwen

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  1. Thomas Erlebach
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  2. Erik Jan van Leeuwen
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Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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Erlebach, T., van Leeuwen, E.J. (2010). PTAS for Weighted Set Cover on Unit Squares. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_13

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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

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