Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers

Authors Sara Ahmadian, Chaitanya Swamy



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2016.69.pdf
  • Filesize: 0.73 MB
  • 15 pages

Document Identifiers

Author Details

Sara Ahmadian
Chaitanya Swamy

Cite AsGet BibTex

Sara Ahmadian and Chaitanya Swamy. Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.69

Abstract

We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| <=k, |sigma^{-1}(i)| >= L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
Keywords
  • Approximation algorithms
  • facililty-location problems
  • primal-dual method
  • Lagrangian relaxation
  • k-center problems
  • minimizing sum of radii

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Gagan Aggarwal, Tomás Feder, Krishnaram Kenthapadi, Samir Khuller, Rina Panigrahy, Dilys Thomas, and An Zhu. Achieving anonymity via clustering. ACM Transactions on Algorithms (TALG), 6(3):49, 2010. Google Scholar
  2. Gagan Aggarwal, Tomás Feder, Krishnaram Kenthapadi, Rajeev Motwani, Rina Panigrahy, Dilys Thomas, and An Zhu. Approximation algorithms for k-anonymity. Journal of Privacy Technology (JOPT), 2005. Google Scholar
  3. Sara Ahmadian and Chaitanya Swamy. Improved approximation guarantees for lower-bounded facility location. In Proceedings of the 10th International Workshop on Approximation and Online Algorithms, pages 257-271. Springer, 2012. Google Scholar
  4. Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. Mathematical Programming, 154(1-2):29-53, 2015. Google Scholar
  5. Babak Behsaz and Mohammad R Salavatipour. On minimum sum of radii and diameters clustering. Algorithmica, 73(1):143-165, 2015. Google Scholar
  6. Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 737-756. SIAM, 2015. Google Scholar
  7. Vasilis Capoyleas, Günter Rote, and Gerhard Woeginger. Geometric clusterings. Journal of Algorithms, 12(2):341-356, 1991. Google Scholar
  8. Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing, 34(4):803-824, 2005. Google Scholar
  9. Moses Charikar, Sudipto Guha, Éva Tardos, and David B Shmoys. A constant-factor approximation algorithm for the k-median problem. In Proceedings of the 31st annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 1999. Google Scholar
  10. Moses Charikar, Samir Khuller, David Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 642-651. SIAM, 2001. Google Scholar
  11. Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 2001. Google Scholar
  12. Ke Chen. A constant factor approximation algorithm for k-median clustering with outliers. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 826-835. SIAM, 2008. Google Scholar
  13. Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. Lp rounding for k-centers with non-uniform hard capacities. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pages 273-282. IEEE, 2012. Google Scholar
  14. Marek Cygan and Tomasz Kociumaka. Constant factor approximation for capacitated k-center with outliers. In Proc. 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), volume 25 of LIPIcs - Leibniz International Proceedings in Informatics, pages 251-262. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2014.251.
  15. Srinivas R Doddi, Madhav V Marathe, SS Ravi, David Scot Taylor, and Peter Widmayer. Approximation algorithms for clustering to minimize the sum of diameters. In Proceedings of the 11th Scandinavian Workshop on Algorithm Theory, pages 237-250, 2000. Google Scholar
  16. Alina Ene, Sariel Har-Peled, and Benjamin Raichel. Fast clustering with lower bounds: No customer too far, no shop too small. arXiv preprint arXiv:1304.7318, 2013. Google Scholar
  17. Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A Pirwani, and Kasturi Varadarajan. On metric clustering to minimize the sum of radii. Algorithmica, 57(3):484-498, 2010. Google Scholar
  18. Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A Pirwani, and Kasturi Varadarajan. On clustering to minimize the sum of radii. SIAM Journal on Computing, 41(1):47-60, 2012. Google Scholar
  19. Sudipto Guha, Adam Meyerson, and Kamesh Munagala. Hierarchical placement and network design problems. In Proceedings of 41st Annual IEEE Symposium on Foundations of Computer Science, pages 603-612. IEEE, 2000. Google Scholar
  20. Dorit S Hochbaum and David B Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10(2):180-184, 1985. Google Scholar
  21. Dorit S Hochbaum and David B Shmoys. A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM, 33(3):533-550, 1986. Google Scholar
  22. Kamal Jain and Vijay V Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM, 48(2):274-296, 2001. Google Scholar
  23. David R. Karger and Maria Minkoff. Building steiner trees with incomplete global knowledge. In Proceedings of 41st Annual IEEE Symposium on Foundations of Computer Science, pages 613-623. IEEE, 2000. Google Scholar
  24. Samir Khuller and Yoram J Sussmann. The capacitated k-center problem. SIAM Journal on Discrete Mathematics, 13(3):403-418, 2000. Google Scholar
  25. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 901-910. ACM, 2013. Google Scholar
  26. Andrew Lim, Fan Wang, and Zhou Xu. A transportation problem with minimum quantity commitment. Transportation Science, 40(1):117-129, 2006. Google Scholar
  27. Pitu B Mirchandani and Richard L Francis. Discrete location theory. Wiley-Interscience, 1990. Google Scholar
  28. Pierangela Samarati. Protecting respondents identities in microdata release. IEEE Transactions on Knowledge and Data Engineering, 13(6):1010-1027, 2001. Google Scholar
  29. David B Shmoys. The design and analysis of approximation algorithms. Trends in Optimization: American Mathematical Society Short Course, January 5-6, 2004, Phoeniz, Arizona, 61:85, 2004. Google Scholar
  30. Zoya Svitkina. Lower-bounded facility location. ACM Transactions on Algorithms (TALG), 6(4):69, 2010. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail