Skip to main content
SpringerLink
Log in

Approximation Algorithms for Robust Clustering Problems Using Local Search Techniques

  • Conference paper
  • First Online:
Theory and Applications of Models of Computation (TAMC 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14637))

Included in the following conference series:

  • 133 Accesses

Abstract

In this paper, we explore two robust models for the k-median and k-means problems: the outlier-version (k-MedO/k-MeaO) and the penalty-version (k-MedP/k-MeaP), enabling the marking and elimination of certain points as outliers. In k-MedO/k-MeaO, the count of outliers is restricted by a specified integer, while in k-MedP/k-MeaP, there’s no explicit limit on outlier quantity, yet each outlier incurs a penalty cost.

We introduce a novel approach to evaluate the approximation ratio of local search algorithms for these problems. This involves an adapted clustering method that captures pertinent information about outliers within both local and global optimal solutions. For k-MeaP, we enhance the best-known approximation ratio derived from local search, elevating it from \(25+\varepsilon \) to \(9+\varepsilon \). The best-known approximation ratio for k-MedP is also obtained.

Regarding k-MedO/k-MeaO, only two bi-criteria approximation algorithms based on local search exist. One violates the outlier constraint (limiting outlier count), while the other breaches the cardinality constraint (restricting the number of clusters). We focus on the former algorithm, enhancing its approximation ratios from \(17+\varepsilon \) to \(3+\varepsilon \) for k-MedO and from \(274+\varepsilon \) to \(9+\varepsilon \) for k-MeaO.

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ahmadian, S., Norouzi-Fard, A., Svensson, O., Ward, J.: Better guarantees for k-means and euclidean k-median by primal-dual algorithms. In: Proceedings of the 58th Annual Symposium on Foundations of Computer Science, pp. 61–72 (2017)

    Google Scholar 

  2. 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 

  3. Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 544–562 (2004)

    Article  MathSciNet  Google Scholar 

  4. Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for \(k\)-median, and positive correlation in budgeted optimization, pp. 737–756 (2014)

    Google Scholar 

  5. Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. J. Comput. Syst. Sci. 65(1), 129–149 (2002)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. Charikar, M., Li, S.: A dependent LP-rounding approach for the k-median problem. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) Automata, Languages, and Programming. ICALP 2012, vol. 7391, pp. 194C205. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31594-7_17

  8. Chen, K.: A constant factor approximation algorithm for k-median clustering with outliers. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 826–835 (2008)

    Google Scholar 

  9. Cohen-Addad, V., Gupta, A., Hu, L., et al.: An improved local search algorithm for k-median. In: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics, pp. 1556–1612 (2022)

    Google Scholar 

  10. Cohen-Addad, V., Karthik, C.: Inapproximability of clustering in \(l_p\) metrics. In: Proceedings of the 60th Annual Symposium on Foundations of Computer Science, pp. 519–539 (2019)

    Google Scholar 

  11. Cohen-Addad, V., Klein, P.N., Mathieu, C.: Local search yields approximation schemes for k-means and k-median in Euclidean and minor-free metrics. SIAM J. Comput. 48(2), 644–667 (2019)

    Google Scholar 

  12. Feng, Q., Zhang, Z., Shi, F., Wang, J.: An improved approximation algorithm for the \(k\)-means problem with penalties. In: Chen, Y., Deng, X., Lu, M. (eds.) Frontiers in Algorithmics. FAW 2019. FAW 2019, vol. 11458, pp. 170–181. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-18126-0_15

  13. Friggstad, Z., Khodamoradi, K., Rezapour, M., Salavatipour, M.R.: Approximation schemes for clustering with outliers. ACM Trans. Algorithms 15(2), 1–26 (2019)

    Article  MathSciNet  Google Scholar 

  14. Friggstad, Z., Rezapour, M., Salavatipour, M.R.: Local search yields a PTAS for k-means in doubling metrics. SIAM J. Comput. 48(2), 452–480 (2019)

    Article  MathSciNet  Google Scholar 

  15. Gupta, S., Kumar, R., Lu, K., Moseley, B., Vassilvitskii, S.: Local search methods for k-means with outliers. In: Proceedings of the 43rd International Conference on Very Large Data Bases, vol. 10, no. (7), p. 757–768 (2017)

    Google Scholar 

  16. Hajiaghayi, M., Khandekar, R., Kortsarz, G.: Local search algorithms for the red-blue median problem. Algorithmica 63(4), 795–814 (2012)

    Article  MathSciNet  Google Scholar 

  17. Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)

    Article  MathSciNet  Google Scholar 

  18. Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48(2), 274–296 (2001)

    Article  MathSciNet  Google Scholar 

  19. Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k-means clustering. Comput. Geom. 28(2–3), 89–112 (2004)

    Article  MathSciNet  Google Scholar 

  20. Krishnaswamy, R., Li, S., Sandeep, S.: Constant approximation for k-median and \(k\)-means with outliers via iterative rounding. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 646–659 (2018)

    Google Scholar 

  21. Li, S., Svensson, O.: Approximating k-median via pseudo-approximation. SIAM J. Comput. 45(2), 530–547 (2016)

    Article  MathSciNet  Google Scholar 

  22. Makarychev, K., Makarychev, Y., Sviridenko, M., Ward, J.: A bi-criteria approximation algorithm for k-means. In: Proceedings of the 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), and the 20th International Workshop on Randomization and Computation (RANDOM), pp. 14:1–14:20 (2016)

    Google Scholar 

  23. Matousek, J.: On approximate geometric k-clustering. Discrete Comput. Geom. 24(1), 61–84 (2000)

    Article  MathSciNet  Google Scholar 

  24. Zhang, D., Hao, C., Wu, C., Xu, D., Zhang, Z.: Local search approximation algorithms for the k-means problem with penalties. J. Comb. Optim. 37(2), 439–453 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Operations Research and Systems Engineering, College of Science, Tianjin University of Technology, Tianjin, 300384, People’s Republic of China

    Chenchen Wu

  2. Institute for Applied Optimization, Department of Computer Science and Technology, Hefei University, Hefei, People’s Republic of China

    Rolf H. Möhring

  3. The Combinatorial Optimization and Graph Algorithms (COGA) group, Institute for Mathematics, Technical University of Berlin, Berlin, Germany

    Rolf H. Möhring

  4. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, People’s Republic of China

    Yishui Wang

  5. Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Dachuan Xu

  6. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

Authors
  1. Chenchen Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rolf H. Möhring
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yishui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dachuan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dongmei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachuan Xu .

Editor information

Editors and Affiliations

  1. Chinese Academy of Sciences, Beijing, China

    Xujin Chen

  2. The Hong Kong Polytechnic University, Hong Kong, China

    Bo Li

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wu, C., Möhring, R.H., Wang, Y., Xu, D., Zhang, D. (2024). Approximation Algorithms for Robust Clustering Problems Using Local Search Techniques. In: Chen, X., Li, B. (eds) Theory and Applications of Models of Computation. TAMC 2024. Lecture Notes in Computer Science, vol 14637. Springer, Singapore. https://doi.org/10.1007/978-981-97-2340-9_17

Download citation

  • DOI: https://doi.org/10.1007/978-981-97-2340-9_17

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-97-2339-3

  • Online ISBN: 978-981-97-2340-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation