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      An algorithm and a core set result for the weighted euclidean one-center problem

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      Author(s)
      Kumar, P.
      Yıldırım, A. E.
      Date
      2009
      Source Title
      INFORMS Journal on Computing
      Print ISSN
      1091-9856
      Electronic ISSN
      1526-5528
      Publisher
      Institute for Operations Research and the Management Sciences (I N F O R M S)
      Volume
      21
      Issue
      4
      Pages
      614 - 629
      Language
      English
      Type
      Article
      Item Usage Stats
      194
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      205
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      Abstract
      Given a set A of m points in n-dimensional space with corresponding positive weights, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point c A n that minimizes the maximum weighted Euclidean distance from c A to each point in A In this paper, given ε > 0, we propose and analyze an algorithm that computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem. Our algorithm explicitly constructs a small subset X ⊆ A, called an ε-core set of A, for which the optimal solution of the corresponding weighted Euclidean one-center problem is a close approximation to that of A. In addition, we establish that \X\ depends only on ε and on the ratio of the smallest and largest weights, but is independent of the number of points m and the dimension n. This result subsumes and generalizes the previously known core set results for the minimum enclosing ball problem. Our algorithm computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem for A in O(mn\X\) arithmetic operations. Our computational results indicate that the size of the ε-core set computed by the algorithm is, in general, significantly smaller than the theoretical worst-case estimate, which contributes to the efficiency of the algorithm, especially for large-scale instances. We shed some light on the possible reasons for this discrepancy between the theoretical estimate and the practical performance.
      Keywords
      Approximation algorithms
      Core sets
      Minimum enclosing balls
      Weighted euclidean one-center problem
      Permalink
      http://hdl.handle.net/11693/22646
      Published Version (Please cite this version)
      http://dx.doi.org/10.1287/ijoc.1080.0315
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