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Maximum Dispersion and Geometric Maximum Weight Cliques

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Approximation Algorithms for Combinatorial Optimization (APPROX 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1913))

Abstract

We consider geometric instances of the problem of finding a set of k vertices in a complete graph with nonnegative edge weights. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. This problem can be considered as a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number of locations from a discrete set of candidates, such that the average distance between selected locations is maximized. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.

Research partially supported by NSERC.

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References

  1. S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for NP-hard problems. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, pages 284–293, 1995.

    Google Scholar 

  2. Y. Asahiro, R. Hassin, and K. Iwama. Complexity of finding dense subgraphs. Manuscript, 2000.

    Google Scholar 

  3. Y. Asahiro, K. Iwama, H. Tamaki, and T. Tokuyama. Greedily finding a dense graph. In Proceedings of the 5th Scandinavian Workshop on Algorithm Theory (SWAT), volume 1097 of Lecture Notes in Computer Science, pages 136–148. Springer-Verlag, 1996.

    Google Scholar 

  4. B. Chandra and M. M. Halldórsson. Facility dispersion and remote subgraphs. In Proceedings of the 5th Scandinavian Workshop on Algorithm Theory (SWAT), volume 1097 of Lecture Notes in Computer Science, pages 53–65. Springer-Verlag, 1996.

    Google Scholar 

  5. E. Erkut. The discrete p-dispersion problem. European Journal of Operations Research, 46:48–60, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. U. Feige, G. Kortsarz, and D. Peleg. The dense k-subgraph problem. Algorithmica, to appear.

    Google Scholar 

  7. U. Feige and M. Seltser. On the densest k-subgraph problems. Technical ReportCS97-16, http://www.wisdom.weizmann.ac.il, 1997.

  8. R. Hassin, S. Rubinstein, and A. Tamir. Approximation algorithms for maximum dispersion. Operations Research Letters, 21:133–137, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Håstad. Clique is hard to approximate within n1-ε. In Proceedings of the 37th IEEE Annual Symposium on Foundations of Computer Science, pages 627–636, 1996.

    Google Scholar 

  10. G. Kortsarz and D. Peleg. On choosing a dense subgraph. In Proceedings of the 34th IEEE Annual Symposium on Foundations of Computer Science, pages 692–701, Palo Alto, CA, 1993.

    Google Scholar 

  11. F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, New York, NY, 1985.

    Google Scholar 

  12. S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi. Heuristic and special case algorithms for dispersion problems. Operations Research, 42:299–310, 1994.

    MATH  Google Scholar 

  13. A. Srivastav and K. Wolf. Finding dense subgraphs with semidefinite programming. In K. Jansen and J. Rolim, editors, Approximation Algorithms for Combinatorial Optimization (APPROX’ 98), volume 1444 of Lecture Notes in Computer Science, pages 181–191, Aalborg, Denmark, 1998. Springer-Verlag.

    Chapter  Google Scholar 

  14. A. Tamir. Obnoxious facility location in graphs. SIAM Journal on Discrete Mathematics, 4:550–567, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Tamir. Comments on the paper: ‘Heuristic and special case algorithms for dispersion problems’ by, S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi. Operations Research, 46:157–158, 1998.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, TU Berlin, 10623, Berlin, Germany

    Sándor P. Fekete

  2. Department of Computing and Information Science, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    Henk Meijer

Authors
  1. Sándor P. Fekete
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  2. Henk Meijer
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Editor information

Editors and Affiliations

  1. Christian-Albrechts-Universität zu Kiel Institut für Informatik und praktische Mathematik Olshausenstr, 40, 24098, Kiel, Germany

    Klaus Jansen

  2. Computer Science Department, University of Maryland, A.V. Williams Bldg College Park, MD 20742, USA

    Samir Khuller

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© 2000 Springer-Verlag Berlin Heidelberg

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Fekete, S.P., Meijer, H. (2000). Maximum Dispersion and Geometric Maximum Weight Cliques. In: Jansen, K., Khuller, S. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2000. Lecture Notes in Computer Science, vol 1913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44436-X_14

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  • DOI: https://doi.org/10.1007/3-540-44436-X_14

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67996-7

  • Online ISBN: 978-3-540-44436-7

  • eBook Packages: Springer Book Archive

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