Skip to main content
SpringerLink
Log in

Approximation Algorithms for Clustering to Minimize the Sum of Diameters

  • Conference paper
  • First Online:
Algorithm Theory - SWAT 2000 (SWAT 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1851))

Included in the following conference series:

Abstract

We consider the problem of partitioning the nodes of a complete edge weighted graph into k clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the development of good approximation algorithms. When edge weights satisfy the triangle inequality, we present the first approximation algorithm for the problem. The approximation algorithm yields a solution that has no more than 10k clusters such that the total diameter of these clusters is within a factor O(log (n/k)) of the optimal value for k clusters, where n is the number of nodes in the complete graph. For any fixed k, we present an approximation algorithm that produces k clusters whose total diameter is at most twice the optimal value. When the distances are not required to satisfy the triangle inequality, we show that, unless P = NP, for any n > 1, there is no polynomial time approximation algorithm that can provide a performance guarantee of n even when the number of clusters is fixed at 3. Other results obtained include a polynomial time algorithm for the problem when the underlying graph is a tree with edge weights.

Research Supported by Department of Energy Contract W-7405-ENG-36 and by NSF Grant CCR-97-34936.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Awerbuch, B. Berger, L. Cowen and D. Peleg, “Near-Linear Time Construction of Sparse Neighborhood Covers”, SI AM J. Computing, Vol. 28, No. 1, 1998, pp. 263–277.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. K. Agarwal and C. M. Procopiuc, “Exact and Approximate Algorithms for Clustering”, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms (SODA’98), San Francisco, CA, Jan. 1998, pp. 658–667.

    Google Scholar 

  3. P. K. Agarwal and C. M. Procopiuc, “Approximation Algorithms for Projective Clustering”, Proc. 11th ACM-SIAM Symposium on Discrete Algorithms (SODA’2000), San Francisco, CA, Jan. 2000, pp. 538–547.

    Google Scholar 

  4. V. Batagelj, S. Korenjak-Cerne and S. Klavzar, “Dynamic Programming and Convex Clustering”, Algorithmica, Vol. 11, No. 2, Feb. 1994, pp. 93–103.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Brucker, “On the Complexity of Clustering Problems”, in Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems, Vol. 157, Edited by M. Beckmann and H. Kunzi, Springer-Verlag, Heidelberg, 1978, pp.45–54.

    Google Scholar 

  6. M. Charikar, C. Chekuri, T. Feder and R. Motwani, “Incremental Clustering and Dynamic Information Retrieval”, Proc. 29th Annual ACM Symposium on Theory of Computing (STOC’97), El Paso, TX, May 1997, pp.626–634.

    Google Scholar 

  7. V. Capoyleas, G. Rote and G. Woeginger, “Geometric Clusterings”, J. Algorithms, Vol. 12, No. 2, Jun. 1991, pp. 341–356.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Datta, “Efficient Parallel Algorithms for Geometric k-Clustering Problems”, Proc. 11th Annual Symposium on Theoretical Aspects of Computer Science (STACS’94), Caen, France, Feb. 1994, Springer-Verlag Lecture Notes in Computer Science, Vol. 775, pp. 475–486.

    Google Scholar 

  9. R. Duda and P. Hart, Pattern Classification and Scene Analysis, Wiley-Interscience, New York, NY, 1973.

    MATH  Google Scholar 

  10. J. S. Deogan, D. Kratsch and G. Steiner, “An Approximation Algorithm for Clustering Graphs with a Dominating Diametral Path”, Information Processing Letters, Vol. 61, No.3, Feb. 1997, pp. 121–127.

    Article  MathSciNet  Google Scholar 

  11. T. Feder and D. H. Greene, “Optimal Algorithms for Approximate Clustering”, Proc. 20th Annual ACM Symposium on Theory of Computing (STOC’88), Chicago, IL, May 1988, pp. 434–444.

    Google Scholar 

  12. R. Fowler, M. Paterson and S. Tanimoto, “Optimal Packing and Covering in the Plane”, Information Processing Letters, Vol. 12, 1981, pp. 133–137.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Guttmann-Beck and R. Hassin, “Approximation Algorithms for Min-sum p-Clustering”, Discrete Applied Mathematics, Vol. 89, 1998, pp. 125–142.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman and Co., San Francisco, CA, 1979.

    MATH  Google Scholar 

  15. T. F. Gonzalez, “Clustering to Minimize the Maximum Intercluster Distance”, Theoretical Computer Science, Vol.38, No. 2-3, Jun. 1985, pp.293–306.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Hansen and B. Jaumard, “Minimum Sum of Diameters Clustering,” Journal of Classification, Vol. 4, 1987, pp. 215–226.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Hansen and B. Jaumard, “Cluster Analysis and Mathematical Programming”, Mathematical Programming, Vol. 79, Aug. 1997, pp. 191–215.

    Google Scholar 

  18. D. S. Hochbaum (Editor), Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, Boston, MA, 1997.

    Google Scholar 

  19. D. S. Hochbaum and D. B. Shmoys, “A Unified Approach to Approximation Algorithms for Bottleneck Problems”, J. ACM, Vol. 33, No. 3, July 1986, pp. 533–550.

    Article  MathSciNet  Google Scholar 

  20. A. Jain and R. Dubes, Algorithms for Clustering Data, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1988.

    MATH  Google Scholar 

  21. S. Khuller, A. Moss and J. Naor, “The Budgeted Maximum Coverage Problem”, Information Processing Letters, Vol. 70, 1999, pp. 39–45.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Matousek, “On Approximate Geometric k-Clustering”, Manuscript, Department of Applied Mathematics, Charles University, Prague, Czech Republic, 1999.

    Google Scholar 

  23. M. V. Marathe, R. Ravi,. R. Sundaram, S. S. Ravi, D. J. Rosenkrantz and H. B. Hunt III,“Bicriteria Network Design Problems”, J. Algorithms, Vol. 28, No. 1, July 1998, pp. 142–171.

    Article  MATH  MathSciNet  Google Scholar 

  24. N. Meggiddo and K. J. Supowit, “On the complexity of some common geometric location problems,” SIAM J. Computing, Vol. 13, 1984, pp. 182–196.

    Article  Google Scholar 

  25. C. L. Monma and S. Suri, “Partitioning Points and Graphs to Minimize the Maximum or the Sum of Diameters”, Proc. 6th Int. Conf. Theory and Applications of Graphs, Kalamazoo, Michigan, May 1989.

    Google Scholar 

  26. J. Plesník, “Complexity of Decomposing Graphs into Factors with Given Diameters or Radii”, Math. Slovaca, Vol. 32, No. 4, 1982, pp. 379–388.

    MATH  MathSciNet  Google Scholar 

  27. U. Pferschy, R. Rudolf and G. J. Woeginger, “Some Geometric Clustering Problems”, Nordic J. Computing, Vol. 1, No. 2, Summer 1994, pp. 246–263.

    MathSciNet  Google Scholar 

  28. P. Raghavan, “Information Retrieval Algorithms: A Survey”, Proc. 8th ACM-SIAM Symposium on Discrete Algorithms (SODA’97), Jan. 1997, pp. 11–18.

    Google Scholar 

  29. T. Zhang, R. Ramakrishnan and M. Livny, “Birch: An Efficient Data Clustering Method for Very Large Databases”, Proc. ACM-SIGMOD International Conference on Management of Data (SIGMOD’96), Aug. 1996, pp. 103–114.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Los Alamos National Laboratory, P. O. Box 1663, MS B265, Los Alamos, NM, 87545, USA

    Srinivas R. Doddi

  2. Los Alamos National Laboratory, P. O. Box 1663, MS M997, Los Alamos, NM, 87545, USA

    Madhav V. Marathe

  3. Department of Computer Science, University at Albany - State University of New York, Albany, NY, 12222, USA

    S. S. Ravi

  4. Department of Computer Science, University of California, Los Angeles, CA, 90095-1596, USA

    David Scot Taylor

  5. Institute for Theoretical Computer Science ETH, 8092, Zürich, Switzerland

    Peter Widmayer

Authors
  1. Srinivas R. Doddi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Madhav V. Marathe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. S. Ravi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David Scot Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peter Widmayer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Doddi, S.R., Marathe, M.V., Ravi, S.S., Taylor, D.S., Widmayer, P. (2000). Approximation Algorithms for Clustering to Minimize the Sum of Diameters. In: Algorithm Theory - SWAT 2000. SWAT 2000. Lecture Notes in Computer Science, vol 1851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44985-X_22

Download citation

  • DOI: https://doi.org/10.1007/3-540-44985-X_22

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67690-4

  • Online ISBN: 978-3-540-44985-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation