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Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors Vincent Cohen-Addad , Karthik C. S. , David Saulpic , Chris Schwiegelshohn

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Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Computational geometry
Keywords
  • k-means clustering
  • k-median clustering
  • Euclidean space
  • Fine-Grained Complexity

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Abstract

The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points).
We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.

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Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.34

Author Details

Vincent Cohen-Addad
  • Google Research, New York, NY, USA
Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
David Saulpic
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
Chris Schwiegelshohn
  • Aarhus University, Denmark

Funding

  • Karthik C. S.: This work was supported by the National Science Foundation under Grants CCF-2313372 and CCF-2443697, a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen, and partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT, Sofia University "St. Kliment Ohridski" as part of the Bulgarian National Roadmap for Research Infrastructure.
  • Schwiegelshohn, Chris: This work was partially supported by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B and by a Google Research Award.

References

  1. 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, Santa Cruz, CA, USA, November 6-9, 2023, pages 1377-1399. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00085.
  2. Enver Aman, Karthik C. S., and Sharath Punna. On connections between k-coloring and Euclidean k-means. In Timothy M. Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms, ESA 2024, September 2-4, 2024, Royal Holloway, London, United Kingdom, volume 308 of LIPIcs, pages 9:1-9:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.9.
  3. Sanjeev Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 2-11. IEEE Computer Society, 1996. URL: https://doi.org/10.1109/SFCS.1996.548458.
  4. Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45(5):753-782, 1998. URL: https://doi.org/10.1145/290179.290180.
  5. Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for Euclidean k-medians and related problems. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 106-113. ACM, 1998. URL: https://doi.org/10.1145/276698.276718.
  6. Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of Euclidean k-means. In Lars Arge and János Pach, editors, 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, volume 34 of LIPIcs, pages 754-767. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.SOCG.2015.754.
  7. Yair Bartal and Lee-Ad Gottlieb. A linear time approximation scheme for Euclidean TSP. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 698-706. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.80.
  8. Vladimir Braverman, Gereon Frahling, Harry Lang, Christian Sohler, and Lin F. Yang. Clustering high dimensional dynamic data streams. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages 576-585. PMLR, 2017. URL: http://proceedings.mlr.press/v70/braverman17a.html.
  9. Moses Charikar, Vincent Cohen-Addad, Ruiquan Gao, Fabrizio Grandoni, Euiwoong Lee, and Ernest van Wijland. An improved greedy approximation for (metric) k-means. In 66th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2025, Sydney, Australia, December 14-17, 2025, pages 233-240. IEEE, 2025. URL: https://doi.org/10.1109/FOCS63196.2025.00016.
  10. Vincent Cohen-Addad. A fast approximation scheme for low-dimensional k-means. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 430-440. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.29.
  11. Vincent Cohen-Addad, Arnaud de Mesmay, Eva Rotenberg, and Alan Roytman. The bane of low-dimensionality clustering. In Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2018. URL: https://doi.org/10.1137/1.9781611975031.30.
  12. Vincent Cohen-Addad, Alessandro Epasto, Silvio Lattanzi, Vahab Mirrokni, Andres Muñoz Medina, David Saulpic, Chris Schwiegelshohn, and Sergei Vassilvitskii. Scalable differentially private clustering via hierarchically separated trees. In Aidong Zhang and Huzefa Rangwala, editors, KDD '22: The 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, August 14 - 18, 2022, pages 221-230. ACM, 2022. URL: https://doi.org/10.1145/3534678.3539409.
  13. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. J. ACM, 68(6):44:1-44:34, 2021. URL: https://doi.org/10.1145/3477541.
  14. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. In J. ACM, volume 68, 2021. URL: https://doi.org/10.1145/3477541.
  15. Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee, Chris Schwiegelshohn, and Ola Svensson. A (2+ε)-approximation algorithm for metric k-median. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 615-624. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718299.
  16. Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight FPT approximations for k-median and k-means. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 42:1-42:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.42.
  17. Vincent Cohen-Addad and Karthik C. S. Inapproximability of clustering in lp metrics. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 519-539. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00040.
  18. Vincent Cohen-Addad, Karthik C. S., and Euiwoong Lee. On approximability of clustering problems without candidate centers. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2635-2648. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.156.
  19. Vincent Cohen-Addad, Karthik C. S., and Euiwoong Lee. Johnson coverage hypothesis: Inapproximability of k-means and k-median in 𝓁_p-metrics. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1493-1530. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.63.
  20. Vincent Cohen-Addad, Philip N. Klein, and Claire Mathieu. 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. URL: https://doi.org/10.1137/17M112717X.
  21. Vincent Cohen-Addad, Silvio Lattanzi, Ashkan Norouzi-Fard, Christian Sohler, and Ola Svensson. Fast and accurate k-means++ via rejection sampling. In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/babcff88f8be8c4795bd6f0f8cccca61-Abstract.html.
  22. Vincent Cohen-Addad, David Saulpic, and Chris Schwiegelshohn. A new coreset framework for clustering. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 169-182. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451022.
  23. Sanjoy Dasgupta and Yoav Freund. Random projection trees for vector quantization. IEEE Transactions on Information Theory, 55(7):3229-3242, 2009. URL: https://doi.org/10.1109/TIT.2009.2021326.
  24. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for exponential-time-hypothesis-tight algorithms and lower bounds in geometric intersection graphs. SIAM J. Comput., 49(6):1291-1331, 2020. URL: https://doi.org/10.1137/20M1320870.
  25. Irit Dinur. Mildly exponential reduction from gap 3sat to polynomial-gap label-cover. Electron. Colloquium Comput. Complex., TR16-128, 2016. URL: https://eccc.weizmann.ac.il/report/2016/128.
  26. Fedor V. Fomin, Petr A. Golovach, and Kirill Simonov. Parameterized k-clustering: Tractability island. J. Comput. Syst. Sci., 117:50-74, 2021. URL: https://doi.org/10.1016/J.JCSS.2020.10.005.
  27. Zachary Friggstad, Mohsen Rezapour, and Mohammad R. Salavatipour. Local search yields a PTAS for k-means in doubling metrics. SIAM J. Comput., 48(2):452-480, 2019. URL: https://doi.org/10.1137/17M1127181.
  28. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. J. Algorithms, 31(1):228-248, 1999. URL: https://doi.org/10.1006/jagm.1998.0993.
  29. S Louis Hakimi. Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Operations research, 13(3):462-475, 1965. Google Scholar
  30. Sariel Har-Peled. Geometric approximation algorithms. Number 173 in Mathematical Surveys and Monographs. American Mathematical Soc., 2011. Google Scholar
  31. Mary Inaba, Naoki Katoh, and Hiroshi Imai. Applications of weighted voronoi diagrams and randomization to variance-based k-clustering (extended abstract). In Proceedings of the Tenth Annual Symposium on Computational Geometry, Stony Brook, New York, USA, June 6-8, 1994, pages 332-339, 1994. URL: https://doi.org/10.1145/177424.178042.
  32. Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 731-740, 2002. URL: https://doi.org/10.1145/509907.510012.
  33. Kamal Jain and Vijay V. Vazirani. 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. URL: https://doi.org/10.1145/375827.375845.
  34. Sándor Kisfaludi-Bak, Jesper Nederlof, and Karol Wegrzycki. A gap-ETH-tight approximation scheme for Euclidean TSP. Journal of the ACM, 72(6):1-48, 2025. Google Scholar
  35. Stavros G. Kolliopoulos and Satish Rao. A nearly linear-time approximation scheme for the Euclidean k-median problem. SIAM J. Comput., 37(3):757-782, 2007. URL: https://doi.org/10.1137/S0097539702404055.
  36. Alfred A Kuehn and Michael J Hamburger. A heuristic program for locating warehouses. Management science, 9(4):643-666, 1963. Google Scholar
  37. Euiwoong Lee, Melanie Schmidt, and John Wright. Improved and simplified inapproximability for k-means. Inf. Process. Lett., 120:40-43, 2017. URL: https://doi.org/10.1016/j.ipl.2016.11.009.
  38. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. SIAM J. Comput., 45(2):530-547, 2016. URL: https://doi.org/10.1137/130938645.
  39. Stuart Lloyd. Least squares quantization in pcm. IEEE transactions on information theory, 28(2):129-137, 1982. URL: https://doi.org/10.1109/TIT.1982.1056489.
  40. Meena Mahajan, Prajakta Nimbhorkar, and Kasturi R. Varadarajan. The planar k-means problem is np-hard. Theor. Comput. Sci., 442:13-21, 2012. URL: https://doi.org/10.1016/J.TCS.2010.05.034.
  41. Konstantin Makarychev, Yury Makarychev, and Ilya P. Razenshteyn. Performance of johnson-lindenstrauss transform for k-means and k-medians clustering. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 1027-1038, 2019. URL: https://doi.org/10.1145/3313276.3316350.
  42. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense csps. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 78:1-78:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.78.
  43. Jirí Matousek. On approximate geometric k-clustering. Discret. Comput. Geom., 24(1):61-84, 2000. URL: https://doi.org/10.1007/s004540010019.
  44. Nimrod Megiddo and Kenneth J Supowit. On the complexity of some common geometric location problems. SIAM journal on computing, 13(1):182-196, 1984. URL: https://doi.org/10.1137/0213014.
  45. Satish B Rao and Warren D Smith. Approximating geometrical graphs via "spanners" and "banyans". In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 540-550, 1998. Google Scholar
  46. John Fred Stollsteimer. The effect of technical change and output expansion on the optimum number, size, and location of pear marketing facilities in a California pear producing region. University of California, Berkeley, 1961. Google Scholar
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