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Clustering in Varying Metrics

Authors Deeparnab Chakrabarty, Jonathan Conroy , Ankita Sarkar

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Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Fixed parameter tractability
Keywords
  • Clustering
  • approximation algorithms
  • LP rounding
  • parameterized and exact algorithms
  • dynamic programming
  • fixed parameter tractability
  • hardness of approximation

Metrics

Abstract

We introduce the aggregated clustering problem, where one is given T instances of a center-based clustering task over the same n points, but under different metrics. The goal is to open k centers to minimize an aggregate of the clustering costs - e.g., the average or maximum - where the cost is measured via k-center/median/means objectives. More generally, we minimize a norm Ψ over the T cost values. We show that for T ≥ 3, the problem is inapproximable to any finite factor in polynomial time. For T = 2, we give constant-factor approximations. We also show W[2]-hardness when parameterized by k, but obtain f(k,T)poly(n)-time 3-approximations when parameterized by both k and T.
When the metrics have structure, we obtain efficient parameterized approximation schemes (EPAS). If all T metrics have bounded ε-scatter dimension, we achieve a (1+ε)-approximation in f(k,T,ε)poly(n) time. If the metrics are induced by edge weights on a common graph G of bounded treewidth tw, and Ψ is the sum function, we get an EPAS in f(T,ε,tw)poly(n,k) time. Conversely, unless (randomized) ETH is false, any finite factor approximation is impossible if parametrized by only T, even when the treewidth is tw = Ω(polylog n).

Cite As Get BibTex

Deeparnab Chakrabarty, Jonathan Conroy, and Ankita Sarkar. Clustering in Varying Metrics. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.19

Author Details

Deeparnab Chakrabarty
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jonathan Conroy
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Ankita Sarkar
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA

Funding

  • Chakrabarty, Deeparnab: Work partially funded by NSF grants CCF-2041920 and CCF-2402571.
  • Conroy, Jonathan: Work partially funded by the U.S. National Science Foundation CAREER Award under the Grant No. CCF-2443017.
  • Sarkar, Ankita: Work partially funded by NSF grants CCF-2041920 and CCF-2402571.

References

  1. Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase. Parameterized approximation schemes for clustering with general norm objectives. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1377-1399. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00085.
  2. Ittai Abraham, Daniel Delling, Amos Fiat, Andrew V Goldberg, and Renato F Werneck. Highway dimension and provably efficient shortest path algorithms. Journal of the ACM (JACM), 63(5):1-26, 2016. URL: https://doi.org/10.1145/2985473.
  3. Ittai Abraham, Amos Fiat, Andrew V Goldberg, and Renato F Werneck. Highway dimension, shortest paths, and provably efficient algorithms. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 782-793. SIAM, 2010. Google Scholar
  4. Akanksha Agrawal, Tanmay Inamdar, Saket Saurabh, and Jie Xue. Clustering what matters: Optimal approximation for clustering with outliers. J. Artif. Int. Res., 78, 2024. Google Scholar
  5. Barbara Anthony, Vineet Goyal, Anupam Gupta, and Viswanath Nagarajan. A plant location guide for the unsure: Approximation algorithms for min-max location problems. Mathematics of Operations Research, 35(1):79-101, 2010. Preliminary version appeared in SODA 2008. URL: https://doi.org/10.1287/MOOR.1090.0428.
  6. Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for euclidean k-medians and related problems. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 106-113, 1998. URL: https://doi.org/10.1145/276698.276718.
  7. Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing (SICOMP), 33(3):544-562, 2004. URL: https://doi.org/10.1137/S0097539702416402.
  8. MohammadHossein Bateni, Hossein Esfandiari, Hendrik Fichtenberger, Monika Henzinger, Rajesh Jayaram, Vahab Mirrokni, and Andreas Wiese. Optimal fully dynamic k-center clustering for adaptive and oblivious adversaries. In the Symposium on Discrete Algorithms (SODA), pages 2677-2727, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH101.
  9. Sayan Bhattacharya, Martín Costa, and Ermiya Farokhnejad. Fully dynamic k-median with near-optimal update time and recourse. In the Symposium on Theory of Computing (STOC), pages 1166-1177, 2025. URL: https://doi.org/10.1145/3717823.3718176.
  10. Sayan Bhattacharya, Martín Costa, Silvio Lattanzi, and Nikos Parotsidis. Fully dynamic k-clustering in Õ(k update time. Adv. in Neural Infomation Processing Systems (NeurIPS), 36:18633-18658, 2023. Google Scholar
  11. Sayan Bhattacharya, Martín Costa, Silvio Lattanzi, and Nikos Parotsidis. Fully dynamic k-center clustering made simple. In the International Conference on Machine Learning (ICML), 2025. Google Scholar
  12. Sujoy Bhore, Ameet Gadekar, and Tanmay Inamdar. Coreset strikes back: Improved parameterized approximation schemes for (constrained) k-median/means. arXiv preprint arXiv:2504.06980, 2025. URL: https://doi.org/10.48550/arXiv.2504.06980.
  13. Romain Bourneuf and Marcin Pilipczuk. Bounding ε-scatter dimension via metric sparsity. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3155-3171. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.101.
  14. Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In the Symposium on Discrete Algorithms (SODA), pages 737-756, 2014. Google Scholar
  15. Constantin Carathéodory. Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen. Rendiconti Del Circolo Matematico di Palermo (1884-1940), 32(1):193-217, 1911. Google Scholar
  16. Deeparnab Chakrabarty, Jonathan Conroy, and Ankita Sarkar. Clustering in varying metrics, 2025. URL: https://arxiv.org/abs/2510.07860.
  17. Deeparnab Chakrabarty, Luc Cote, and Ankita Sarkar. Fault-tolerant k-supplier with outliers. In the Symposium on Theoretical Aspects of Computer Science (STACS). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.23.
  18. Deeparnab Chakrabarty, Prachi Goyal, and Ravishankar Krishnaswamy. The Non-Uniform k-Center Problem. the ACM Transactions on Algorithms (TALG), 2020. Preliminary version in ICALP 2016. URL: https://doi.org/10.1145/3392720.
  19. Deeparnab Chakrabarty and Maryam Negahbani. Generalized Center Problems with Outliers. the ACM Transactions on Algorithms (TALG), 2019. Prelim. version in ICALP 2018. URL: https://doi.org/10.1145/3338513.
  20. Deeparnab Chakrabarty, Maryam Negahbani, and Ankita Sarkar. Approximation algorithms for continuous clustering and facility location problems. In European Symposium on Algorithms, 2022. Google Scholar
  21. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In the Symposium on Theory of Computing (STOC), pages 126-137, 2019. URL: https://doi.org/10.1145/3313276.3316322.
  22. TH Hubert Chan, Arnaud Guerqin, and Mauro Sozio. Fully dynamic k-center clustering. In the International World Wide Web Conference, pages 579-587, 2018. Google Scholar
  23. Moses Charikar, Sudipto Guha, Éva Tardos, and David B Shmoys. A constant-factor approximation algorithm for the k-median problem. Journal of Computer and System Sciences, 65(1):129-149, 2002. Preliminary version appeared in STOC 1999. URL: https://doi.org/10.1006/JCSS.2002.1882.
  24. Moses Charikar, Samir Khuller, David M Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In the Symposium on Discrete Algorithms (SODA), volume 1, pages 642-651. Citeseer, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365555.
  25. Moses Charikar and Shi Li. A dependent lp-rounding approach for the k-median problem. In International Colloquium on Automata, Languages, and Programming, pages 194-205. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_17.
  26. Eden Chlamtáč, Yury Makarychev, and Ali Vakilian. Approximating fair clustering with cascaded norm objectives. In the Symposium on Discrete Algorithms (SODA), 2022. Google Scholar
  27. Fabián A Chudak and David B Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. Journal of the ACM (JACM), 33(1):1-25, 2003. URL: https://doi.org/10.1137/S0097539703405754.
  28. Julia Chuzhoy and Zihan Tan. Towards tight (er) bounds for the excluded grid theorem. Journal of Combinatorial Theory, Series B, 146:219-265, 2021. URL: https://doi.org/10.1016/J.JCTB.2020.09.010.
  29. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. Journal of the ACM (JACM), 68(6):1-34, 2021. URL: https://doi.org/10.1145/3477541.
  30. 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. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.42.
  31. Vincent Cohen-Addad, Niklas Oskar D Hjuler, Nikos Parotsidis, David Saulpic, and Chris Schwiegelshohn. Fully dynamic consistent facility location. Adv. in Neural Infomation Processing Systems (NeurIPS), 32, 2019. Google Scholar
  32. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  33. Shichuan Deng, Jian Li, and Yuval Rabani. Approximation algorithms for clustering with dynamic points. Journal of Computer and System Sciences, 130:43-70, 2022. URL: https://doi.org/10.1016/J.JCSS.2022.07.001.
  34. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  35. Dan Feldman and Michael Langberg. A unified framework for approximating and clustering data. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 569-578, 2011. URL: https://doi.org/10.1145/1993636.1993712.
  36. Hendrik Fichtenberger, Silvio Lattanzi, Ashkan Norouzi-Fard, and Ola Svensson. Consistent k-clustering for general metrics. In the Symposium on Discrete Algorithms (SODA), pages 2660-2678, 2021. URL: https://doi.org/10.1137/1.9781611976465.158.
  37. Sebastian Forster and Antonis Skarlatos. Dynamic consistent k-center clustering with optimal recourse. In the Symposium on Discrete Algorithms (SODA), pages 212-254, 2025. URL: https://doi.org/10.1137/1.9781611978322.7.
  38. Ameet Gadekar and Tanmay Inamdar. Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering. In Olaf Beyersdorff, Michał Pilipczuk, Elaine Pimentel, and Nguyên Kim Thang, editors, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025), volume 327 of Leibniz International Proceedings in Informatics (LIPIcs), pages 35:1-35:20, Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2025.35.
  39. Dorit S Hochbaum and David B Shmoys. A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM (JACM), 33(3):533-550, 1986. URL: https://doi.org/10.1145/5925.5933.
  40. AJ Hoffman and JB Kruskal. Integral boundary points of convex polyhedra, in “linear inequalities and related systems”(hw kuhn and aw tucker, eds.). Annals of Mathematical Studies, 38, 1956. Google Scholar
  41. Kamal Jain and Vijay V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangean relaxation. Journal of the ACM (JACM), 48(2):274-296, 2001. URL: https://doi.org/10.1145/375827.375845.
  42. Stavros G Kolliopoulos and Satish Rao. A nearly linear-time approximation scheme for the euclidean k-median problem. SIAM Journal on Computing, 37(3):757-782, 2007. URL: https://doi.org/10.1137/S0097539702404055.
  43. Ravishankar Krishnaswamy, Shi Li, and Sai Sandeep. Constant approximation for k-median and k-means with outliers via iterative rounding. In the Symposium on Theory of Computing (STOC), pages 646-659, 2018. URL: https://doi.org/10.1145/3188745.3188882.
  44. Shrinu Kushagra. Three-dimensional matching is np-hard. arXiv preprint arXiv:2003.00336, 2020. URL: https://arxiv.org/abs/2003.00336.
  45. Shrinu Kushagra, Shai Ben-David, and Ihab Ilyas. Semi-supervised clustering for de-duplication. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 1659-1667. PMLR, 2019. URL: http://proceedings.mlr.press/v89/kushagra19a.html.
  46. Jakub Łącki, Bernhard Haeupler, Christoph Grunau, Rajesh Jayaram, and Václav Rozhoň. Fully dynamic consistent k-center clustering. In the Symposium on Discrete Algorithms (SODA), pages 3463-3484, 2024. Google Scholar
  47. Silvio Lattanzi and Sergei Vassilvitskii. Consistent k-clustering. In the International Conference on Machine Learning (ICML), pages 1975-1984, 2017. URL: http://proceedings.mlr.press/v70/lattanzi17a.html.
  48. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222:45-58, 2013. URL: https://doi.org/10.1016/J.IC.2012.01.007.
  49. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. SIAM Journal on Computing (SICOMP), 45(2):530-547, 2016. URL: https://doi.org/10.1137/130938645.
  50. Sepideh Mahabadi and Ali Vakilian. (Individual) Fairness for k-Clustering. In the International Conference on Machine Learning (ICML), pages 7925-7935, 2020. URL: https://dl.acm.org/doi/abs/10.5555/3524938.3525549.
  51. Konstantin Makarychev, Yury Makarychev, and Ilya Razenshteyn. Performance of johnson-lindenstrauss transform for k-means and k-medians clustering. In the Symposium on Theory of Computing (STOC), pages 1027-1038, 2019. URL: https://doi.org/10.1145/3313276.3316350.
  52. Yury Makarychev and Ali Vakilian. Approximation algorithms for socially fair clustering. In the Conference on Learning Theory (COLT), 2021. Google Scholar
  53. 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.
  54. Neil Robertson and Paul D Seymour. Graph minors. v. excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. URL: https://doi.org/10.1016/0095-8956(86)90030-4.
  55. David B Shmoys, Éva Tardos, and Karen Aardal. Approximation algorithms for facility location problems. In the Symposium on Theory of Computing (STOC), pages 265-274, 1997. Google Scholar
  56. Ernst Steinitz. Bedingt konvergente reihen und konvexe systeme. Journal für die reine und angewandte Mathematik, 1913. Google Scholar
  57. Chaitanya Swamy. Improved approximation algorithms for matroid and knapsack median problems and applications. the ACM Transactions on Algorithms (TALG), 12(4):1-22, 2016. URL: https://doi.org/10.1145/2963170.
  58. Li Yan and Marek Chrobak. Lp-rounding algorithms for the fault-tolerant facility placement problem. Journal of Discrete Algorithms, 33:93-114, 2015. URL: https://doi.org/10.1016/J.JDA.2015.03.004.
  59. Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In the International Conference on Machine Learning (ICML), pages 928-936, 2003. URL: http://www.aaai.org/Library/ICML/2003/icml03-120.php.
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