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

Date
2009
Authors
Kumar, P.
Yıldırım, A. E.
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Supervisor
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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
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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.

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Keywords
Approximation algorithms, Core sets, Minimum enclosing balls, Weighted euclidean one-center problem
Citation
Published Version (Please cite this version)