PTAS for geometric hitting set problems via local search

Nabil H. Mustafa, Saurabh Ray

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are halfspaces in ℝ3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.

    Original languageEnglish (US)
    Title of host publicationProceedings of the 25th Annual Symposium on Computational Geometry, SCG'09
    Pages17-22
    Number of pages6
    DOIs
    StatePublished - Dec 4 2009
    Event25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark
    Duration: Jun 8 2009Jun 10 2009

    Other

    Other25th Annual Symposium on Computational Geometry, SCG'09
    CountryDenmark
    CityAarhus
    Period6/8/096/10/09

    Fingerprint

    Hitting Set
    Geometric object
    Local Search
    Approximation algorithms
    Polynomials
    Admissible Set
    Local Search Algorithm
    Hits
    Iterate
    Half-space
    Set of points
    Unit Disk
    Polynomial-time Algorithm
    Approximation Algorithms
    NP-complete problem
    Subset
    Computing

    Keywords

    • Approximation algorithm
    • Epsilon nets
    • Hitting sets
    • Local search

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Geometry and Topology
    • Computational Mathematics

    Cite this

    Mustafa, N. H., & Ray, S. (2009). PTAS for geometric hitting set problems via local search. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 (pp. 17-22) https://doi.org/10.1145/1542362.1542367

    PTAS for geometric hitting set problems via local search. / Mustafa, Nabil H.; Ray, Saurabh.

    Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 17-22.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Mustafa, NH & Ray, S 2009, PTAS for geometric hitting set problems via local search. in Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. pp. 17-22, 25th Annual Symposium on Computational Geometry, SCG'09, Aarhus, Denmark, 6/8/09. https://doi.org/10.1145/1542362.1542367
    Mustafa NH, Ray S. PTAS for geometric hitting set problems via local search. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 17-22 https://doi.org/10.1145/1542362.1542367
    Mustafa, Nabil H. ; Ray, Saurabh. / PTAS for geometric hitting set problems via local search. Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. pp. 17-22
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