Document Open Access Logo

On Tight FPT Time Approximation Algorithms for k-Clustering Problems

Authors Han Dai , Shi Li , Sijin Peng

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.72.pdf
  • Filesize: 1.04 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • Approximation algorithms
  • Monotone symmetric norms
  • Clustering
  • Fixed parameter tractability

Metrics

Abstract

Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm k-clustering problems, parameterized by the number k of open facilities.
For the capacitated setting, we give a tight (3+ε)-approximation for the general-norm capacitated k-clustering problem in FPT-time parameterized by k and ε. Prior to our work, such a result was only known for the capacitated k-median problem [Cohen-Addad and Li, 2019]. As a special case, our result yields an FPT-time 3-approximation for capacitated k-center. The problem has not been studied in the FPT-time setting, with the previous best known polynomial-time approximation ratio being 9 [An et al., 2015]. 
In the uncapacitated setting, we consider the top-cn norm k-clustering problem, where the goal of the problem is to minimize the top-cn norm of the connection distance vector. Our main result is a tight (1 + 2/(ec) + ε)-approximation algorithm for the problem with c ∈ (1/e, 1]. (For the case c ≤ 1/e, there is a simple tight (3+ε)-approximation.) Our framework can be easily extended to give a tight (3, 1 + 2/e + ε)-bi-criteria approximation for the (k-center, k-median) problem in FPT time, improving the previous best polynomial-time (4, 8) guarantee [Soroush Alamdari and David B. Shmoys, 2017]. 
All results are based on a unified framework: computing a (1+ε)-approximate solution using O((k log n)/ε) facilities S via LP rounding, sampling a few client representatives R based on the solution S, guessing a few pivots from S ∪ R and some radius information on the pivots, and solving the problem using the guesses. We believe this framework can lead to further results on k-clustering problems.

Cite As Get BibTex

Han Dai, Shi Li, and Sijin Peng. On Tight FPT Time Approximation Algorithms for k-Clustering Problems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 72:1-72:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.72

Author Details

Han Dai
  • School of Computer Science, Nanjing University, China
Shi Li
  • School of Computer Science, Nanjing University, China
Sijin Peng
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Dai, Han: Supported by State Key Laboratory for Novel Software Technology, and New Cornerstone Science Foundation.
  • Li, Shi: Supported by State Key Laboratory for Novel Software Technology, and New Cornerstone Science Foundation.

References

  1. Karen Aardal, Pieter L.van den Berg, Dion Gijswijt, and Shanfei Li. Approximation algorithms for hard capacitated k-facility location problems. European Journal of Operational Research, 242(2):358-368, 2015. URL: https://doi.org/10.1016/J.EJOR.2014.10.011.
  2. Fateme Abbasi, Sandip Banerjee, Jaroslaw Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase. Parameterized approximation schemes for clustering with general norm objectives. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, pages 1377-1399, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00085.
  3. Marek Adamczyk, Jaroslaw Byrka, Jan Marcinkowski, Syed Mohammad Meesum, and Michal Wlodarczyk. Constant-factor FPT approximation for capacitated k-median. In 27th Annual European Symposium on Algorithms, ESA 2019, volume 144, pages 1:1-1:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.1.
  4. Sara Ahmadian, Ashkan Norouzi-Fard, Ola Svensson, and Justin Ward. Better guarantees for k-means and euclidean k-median by primal-dual algorithms. SIAM Journal on Computing, 49(4), 2020. URL: https://doi.org/10.1137/18M1171321.
  5. Soroush Alamdari and David B. Shmoys. A bicriteria approximation algorithm for the k-center and k-median problems. In Workshop on Approximation and Online Algorithms, WAOA 2017, volume 10787, pages 66-75, 2017. URL: https://doi.org/10.1007/978-3-319-89441-6_6.
  6. 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):29-53, 2015. URL: https://doi.org/10.1007/S10107-014-0857-Y.
  7. Ali Aouad and Danny Segev. The ordered k-median problem: surrogate models and approximation algorithms. Mathematical Programming, 177(1-2):55-83, 2019. URL: https://doi.org/10.1007/S10107-018-1259-3.
  8. Judit Barilan, Guy Kortsarz, and David Peleg. How to allocate network centers. Journal of Algorithms, 15(3):385-415, 1993. URL: https://doi.org/10.1006/JAGM.1993.1047.
  9. Vladimir Braverman, Vincent Cohen-Addad, Shaofeng H.-C. Jiang, Robert Krauthgamer, Chris Schwiegelshohn, Mads Bech Toftrup, and Xuan Wu. The power of uniform sampling for coresets. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022,, pages 462-473, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00051.
  10. Vladimir Braverman, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Xuan Wu. Coresets for ordered weighted clustering. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, volume 97, pages 744-753, 2019. URL: http://proceedings.mlr.press/v97/braverman19a.html.
  11. Jarosław Byrka, Krzysztof Fleszar, Bartosz Rybicki, and Joachim Spoerhase. Bi-factor approximation algorithms for hard capacitated k-median problems. In Proceedings of the 26h Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pages 722-736, 2014. URL: https://doi.org/10.1137/1.9781611973730.49.
  12. Jaroslaw Byrka, Thomas W. Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median and positive correlation in budgeted optimization. ACM Transactions on Algorithms, 13(2):23:1-23:31, 2017. URL: https://doi.org/10.1145/2981561.
  13. Jarosław Byrka, Bartosz Rybicki, and Sumedha Uniyal. An approximation algorithm for uniform capacitated k-median problem with (1+ε)-capacity violation. In International Conference on Integer Programming and Combinatorial Optimization, IPCO 2016, volume 9682, pages 262-274, 2016. URL: https://doi.org/10.1007/978-3-319-33461-5_22.
  14. Jaroslaw Byrka, Krzysztof Sornat, and Joachim Spoerhase. Constant-factor approximation for ordered k-median. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 620-631, 2018. URL: https://doi.org/10.1145/3188745.3188930.
  15. Deeparnab Chakrabarty and Chaitanya Swamy. Interpolating between k-median and k-center: Approximation algorithms for ordered k-median. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, volume 107, pages 29:1-29:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.29.
  16. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 126-137, 2019. URL: https://doi.org/10.1145/3313276.3316322.
  17. Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for the facility location and k-median problems. In 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, pages 378-388, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814609.
  18. Ke Chen. On k-median clustering in high dimensions. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, pages 1177-1185, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109687.
  19. Ke Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009. URL: https://doi.org/10.1137/070699007.
  20. Julia Chuzhoy and Yuval Rabani. Approximating k-median with non-uniform capacities. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, volume 5, pages 952-958, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070569.
  21. Vincent Cohen-Addad, Hossein Esfandiari, Vahab Mirrokni, and Shyam Narayanan. Improved approximations for euclidean k-means and k-median, via nested quasi-independent sets. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 1621-1628, 2022. URL: https://doi.org/10.1145/3519935.3520011.
  22. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. Journal of the ACM, 68(6):1-34, 2021. URL: https://doi.org/10.1145/3477541.
  23. Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee, Chris Schwiegelshohn, and Ola Svensson. A (2+ ε)-approximation algorithm for metric k-median. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, pages 615-624, 2025. URL: https://doi.org/10.1145/3717823.3718299.
  24. Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight fpt approximations for k-median and k-means. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, volume 132, pages 42:1-42:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.42.
  25. Vincent Cohen-Addad and Jason Li. On the fixed-parameter tractability of capacitated clustering. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, volume 132, pages 41-1, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.41.
  26. Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. Lp rounding for k-centers with non-uniform hard capacities. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, pages 273-282, 2012. URL: https://doi.org/10.1109/FOCS.2012.63.
  27. Gökalp Demirci and Shi Li. Constant approximation for capacitated k-median with (1+ ε)-capacity violation. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, volume 55, pages 73:1-73:14, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.73.
  28. David Eisenstat, Philip N. Klein, and Claire Mathieu. Approximating k-center in planar graphs. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 617-627, 2014. URL: https://doi.org/10.1137/1.9781611973402.47.
  29. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, STOC 2003, pages 448-455, 2003. URL: https://doi.org/10.1145/780542.780608.
  30. Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In Proceedings of the 21th Annual ACM Symposium on Theory of Computing, STOC 1988, pages 434-444, 1988. URL: https://doi.org/10.1145/62212.62255.
  31. Dan Feldman, Morteza Monemizadeh, and Christian Sohler. A ptas for k-means clustering based on weak coresets. In Proceedings of the 23rd Annual Symposium on Computational Geometry, SoCG 2007, pages 11-18, 2007. URL: https://doi.org/10.1145/1247069.1247072.
  32. Dan Feldman, Melanie Schmidt, and Christian Sohler. Turning big data into tiny data: Constant-size coresets for k-means, pca, and projective clustering. SIAM Journal on Computing, 49(3):601-657, 2020. URL: https://doi.org/10.1137/18M1209854.
  33. Zachary Friggstad, Mohsen Rezapour, and Mohammad R Salavatipour. Local search yields a ptas for k-means in doubling metrics. SIAM Journal on Computing, 48(2):452-480, 2019. URL: https://doi.org/10.1137/17M1127181.
  34. Teofilo F Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985. URL: https://doi.org/10.1016/0304-3975(85)90224-5.
  35. Dishant Goyal and Ragesh Jaiswal. Tight FPT approximation for constrained k-center and k-supplier. Theoretical Computer Science, 940:190-208, 2023. URL: https://doi.org/10.1016/J.TCS.2022.11.001.
  36. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. Journal of Algorithms, 31(1):228-248, 1999. URL: https://doi.org/10.1006/JAGM.1998.0993.
  37. 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. URL: https://doi.org/10.1287/MOOR.10.2.180.
  38. 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. URL: https://doi.org/10.1145/5925.5933.
  39. Lingxiao Huang, Jian Li, Pinyan Lu, and Xuan Wu. Coresets for constrained clustering: General assignment constraints and improved size bounds. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, pages 4732-4782, 2025. URL: https://doi.org/10.1137/1.9781611978322.161.
  40. Kamal Jain and Vijay V. Vazirani. Primal-dual approximation algorithms for metric facility location and k-median problems. In 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, pages 2-13, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814571.
  41. Samir Khuller and Yoram J Sussmann. The capacitated k-center problem. SIAM Journal on Discrete Mathematics, 13(3):403-418, 2000. URL: https://doi.org/10.1137/S0895480197329776.
  42. Silvio Lattanzi and Christian Sohler. A better k-means++ algorithm via local search. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, volume 97, pages 3662-3671, 2019. URL: http://proceedings.mlr.press/v97/lattanzi19a.html.
  43. Shanfei Li. An improved approximation algorithm for the hard uniform capacitated k-median problem. arXiv preprint arXiv:1406.4454, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.325.
  44. Shi Li. Approximating capacitated k-median with (1+ε)k open facilities. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 786-796, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH56.
  45. Shi Li. On uniform capacitated k-median beyond the natural lp relaxation. ACM Transactions on Algorithms, 13(2):1-18, 2017. URL: https://doi.org/10.1145/2983633.
  46. Michal Osadnik. Fixed parameter tractable algorithm and coreset for the ordered k-median problem, master thesis, 2023. URL: https://urn.fi/URN:NBN:fi:aalto-202405193486.
  47. Ján Plesník. A heuristic for the p-center problems in graphs. Discrete Applied Mathematics, 17(3):263-268, 1987. URL: https://doi.org/10.1016/0166-218X(87)90029-1.
  48. Arie Tamir. The k-centrum multi-facility location problem. Discrete Applied Mathematics, 109(3):293-307, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00253-5.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact