Skip to main content
SpringerLink
Log in

NP-hardness of Some Max-Min Clustering Problems

  • Conference paper
  • First Online:
Book cover Optimization and Applications (OPTIMA 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 974))

Included in the following conference series:

Abstract

We consider some consimilar problems of searching for disjoint clusters in the finite set of points in Euclidean space. The goal is to maximize the minimum subset size so that the value of each intracluster quadratic variation would not exceed a given constant. We prove that all considered problems are NP-hard even on a line.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

References

  1. MacQueen, J.B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297. University of California Press, Berkeley (1967)

    Google Scholar 

  2. Rao, M.: Cluster analysis and mathematical programming. J. Amer. Stat. Assoc. 66, 622–626 (1971)

    Article  Google Scholar 

  3. Hansen, P., Jaumard, B., Mladenovich, N.: Minimum sum of squares clustering in a low dimensional space. J. Classif. 15, 37–55 (1998)

    Article  MathSciNet  Google Scholar 

  4. Hansen, P., Jaumard, B.: Cluster analysis and mathematical programming. Math. Program. 79, 191–215 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-hardness of Euclidean sum-of-squares clustering. Mach. Learn. 75(2), 245–248 (2009)

    Article  Google Scholar 

  6. Drineas, P., Frieze, A., Kannan, R., Vempala, S., Vinay, V.: Clustering large graphs via the singular value decomposition. Mach. Learn. 56, 9–33 (2004)

    Article  Google Scholar 

  7. Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 106–113 (1998)

    Google Scholar 

  8. Kariv, O., Hakimi, S.: An algorithmic approach to network location problems. Part 1: the \(p\)-centers. SIAM J. Appl. Math. 37, 513–538 (1979)

    Article  MathSciNet  Google Scholar 

  9. Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of 20th Annual ACM Symposium on Theory of Computing, pp. 434–444 (1988)

    Google Scholar 

  10. Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the k-center problem. Math. Oper. Res. 10(2), 180–184 (1985)

    Article  MathSciNet  Google Scholar 

  11. Kaufman, L., Rousseeuw, P.J.: Clustering by means of Medoids, in statistical data analysis based on the \(L_{1}\)-Norm and related methods. Edited by Dodge, Y., pp. 405–416. Dodge, North-Holland (1987)

    Google Scholar 

  12. Krarup, J., Pruzan, P.: The simple plant location problem: survey and synthesis. Eur. J. Oper. Res. 12(1), 36–81 (1983)

    Article  MathSciNet  Google Scholar 

  13. Mirchandani, P., Francis, R. (eds.): Discrete Location Theory. Wiley-Intersience, New York (1990)

    MATH  Google Scholar 

  14. Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of 12th ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651 (2001)

    Google Scholar 

  15. Agarwal, P.K., Phillips, J.M.: An efficient algorithm for 2D Euclidean 2-center with outliers. In: Proceedings of 16th Annual European Symposium Algorithms, pp. 64–75 (2008)

    Google Scholar 

  16. McCutchen, R.M., Khuller, S.: Streaming algorithms for k-center clustering with outliers and with anonymity. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX/RANDOM -2008. LNCS, vol. 5171, pp. 165–178. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85363-3_14

    Chapter  MATH  Google Scholar 

  17. Hatami, B., Zarrabi-Zade, H.: A streaming algorithm for 2-center with outliers in high dimensions. Comput. Geom. 60, 26–36 (2017)

    Article  MathSciNet  Google Scholar 

  18. Jain, A.K.: Data clustering: 50 years beyond \(k\)-Means. Patt. Recogn. Lett. 31(8), 651–666 (2010)

    Article  Google Scholar 

  19. Shirkhorshidi, A.S., Aghabozorgi, S., Wah, T.Y., Herawan, T.: Big data clustering: a review. In: Murgante, B., et al. (eds.) ICCSA 2014, Part V. LNCS, vol. 8583, pp. 707–720. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09156-3_49

    Chapter  Google Scholar 

  20. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)

    MATH  Google Scholar 

  21. James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning. Springer, New York (2013)

    Book  Google Scholar 

  22. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York (2009)

    Book  Google Scholar 

  23. Aggarwal, C.C.: Data Mining: The Textbook. Springer, Switzerland (2015)

    MATH  Google Scholar 

  24. Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning (Adaptive Computation and Machine Learning series). The MIT Press (2017)

    Google Scholar 

  25. Fu, T.-C.: A review on time series data mining. Eng. Appl. Artif. Intell. 24(1), 164–181 (2011)

    Article  Google Scholar 

  26. Kuenzer, C., Dech, S., Wagner, W.: Remote Sensing Time Series. Remote Sensing and Digital Image Processing, vol. 22. Springer, Switzerland (2015)

    Google Scholar 

  27. Liao, T.W.: Clustering of time series data – a survey. Patt. Recogn. 38(11), 1857–1874 (2005)

    Article  Google Scholar 

  28. de Waal, T., Pannekoek, J., Scholtus, S.: Handbook of Statistical Data Editing and Imputation. Wiley, New Jersey (2011)

    Book  Google Scholar 

  29. Osborne, J.W.: Best Practices in Data Cleaning: A Complete Guide to Everything You Need to Do Before and After Collecting Your Data, 1st edn. SAGE Publication, Inc., Los Angeles (2013)

    Book  Google Scholar 

  30. Farcomeni, A., Greco, L.: Robust Methods for Data Reduction. Chapman and Hall/CRC (2015)

    Google Scholar 

  31. Ageev, A.A., Kel’manov, A.V., Pyatkin, A.V., Khamidullin, S.A., Shenmaier, V.V.: Approximation polynomial algorithm for the data editing and data cleaning problem. Patt. Recogn. Image Anal. 27(3), 365–370 (2017)

    Article  Google Scholar 

  32. Ageev, A.A., Kel’manov, A.V., Pyatkin, A.V., Khamidullin, S.A., Shenmaier, V.V.: 1/2-Approximation polynomial-time algorithm for a problem of searching a subset. In: Proceedings of 2017 International MultiConference on Engineering, Computer and Information Sciences (SIBIRCON), 18–22 September 2017, Novosibirsk, Russia, pp. 8–12 (2017)

    Google Scholar 

  33. Kel’manov, A.V., Khamidullin, S.A., Khandeev, V.I., Pyatkin, A.V.: Exact algorithms for two quadratic Euclidean problems of searching for the largest subset and longest subsequence. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P.M. (eds.) LION 2018. LNCS, vol. 11353, pp. 308–318. Springer (2018). https://doi.org/10.1007/978-3-030-05348-2_28

    Google Scholar 

  34. Kel’manov, A.V., Khandeev, V.I., Panasenko A.V.: Exact algorithms for the special cases of two hard to solve problems of searching for the largest subset. In: van der Aalst, W.M.P., et al. (eds.) AIST 2018. LNCS, vol. 11179, pp. 294–304. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-11027-7_28

    Google Scholar 

  35. Kel’manov, A.V., Pyatkin, A.V., Khamidullin, S.A., Khandeev, V.I., Shenmaier, V.V., Shamardin, Yu.V.: An approximation polynomial algorithm for a problem of searching for the longest subsequence in a finite sequence of points in Euclidean space. In: Eremeev, A., Khachay, M., Kochetov, Y., Pardalos, P. (eds.) OPTA 2018. CCIS, vol. 871, pp. 120–130. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-93800-4_10

    Google Scholar 

  36. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

Download references

Acknowledgments

The study presented was supported by the Russian Science Foundation, project 16-11-10041.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Koptyug Ave., 630090, Novosibirsk, Russia

    Alexander Kel’manov, Vladimir Khandeev & Artem Pyatkin

  2. Novosibirsk State University, 2 Pirogova St., 630090, Novosibirsk, Russia

    Alexander Kel’manov, Vladimir Khandeev & Artem Pyatkin

Authors
  1. Alexander Kel’manov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Khandeev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Artem Pyatkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Khandeev .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Mikhail Posypkin

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kel’manov, A., Khandeev, V., Pyatkin, A. (2019). NP-hardness of Some Max-Min Clustering Problems. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-10934-9_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10933-2

  • Online ISBN: 978-3-030-10934-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation