Document Open Access Logo

On Approximability of 𝓁₂² Min-Sum Clustering

Authors Karthik C. S. , Euiwoong Lee , Yuval Rabani , Chris Schwiegelshohn , Samson Zhou

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2025.62.pdf
  • Filesize: 0.96 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Computational geometry
  • Theory of computation β†’ Facility location and clustering
Keywords
  • Clustering
  • hardness of approximation
  • polynomial-time approximation schemes
  • learning-augmented algorithms

Metrics

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

Abstract

The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize βˆ‘_{i=1}^k βˆ‘_{p,q ∈ C_i} β€–p-qβ€–β‚‚Β². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. 
In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. 
We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}dβ‹… 2^{(k/Ξ΅)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error Ξ± ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+Ξ³Ξ±)/(1-Ξ±)Β²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant Ξ³ > 0.

Cite As Get BibTex

Karthik C. S., Euiwoong Lee, Yuval Rabani, Chris Schwiegelshohn, and Samson Zhou. On Approximability of 𝓁₂² Min-Sum Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.62

Author Details

Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Yuval Rabani
  • The Hebrew University of Jerusalem, Israel
Chris Schwiegelshohn
  • Aarhus University, Denmark
Samson Zhou
  • Texas A&M University, College Station, TX, USA

Funding

  • Karthik C. S.: Supported by NSF 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.
  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669 and Google.
  • Rabani, Yuval: Supported in part by ISF grants 3565-21 and 389-22, and by BSF grant 2023607.
  • Schwiegelshohn, Chris: Supported in part by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B and Google.
  • Zhou, Samson: Supported in part by NSF CCF-2335411. The work was conducted in part while the author was visiting the Simons Institute for the Theory of Computing as part of the Sublinear Algorithms program.

References

  1. Anders Aamand, Justin Y. Chen, and Piotr Indyk. (optimal) online bipartite matching with degree information. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems, NeurIPS, 2022. Google Scholar
  2. Daniel Aloise, Amit Deshpande, Pierre Hansen, and Preyas Popat. Np-hardness of euclidean sum-of-squares clustering. Mach. Learn., 75(2):245-248, 2009. URL: https://doi.org/10.1007/S10994-009-5103-0.
  3. Enver Aman, Karthik C. S., and Sharath Punna. On connections between k-coloring and Euclidean k-means. In 32nd Annual European Symposium on Algorithms, ESA 2024, 2024. To appear. Google Scholar
  4. Keerti Anand, Rong Ge, Amit Kumar, and Debmalya Panigrahi. Online algorithms with multiple predictions. In International Conference on Machine Learning, ICML, pages 582-598, 2022. URL: https://proceedings.mlr.press/v162/anand22a.html.
  5. Antonios Antoniadis, Christian Coester, Marek EliΓ‘s, Adam Polak, and Bertrand Simon. Online metric algorithms with untrusted predictions. ACM Trans. Algorithms, 19(2):19:1-19:34, 2023. URL: https://doi.org/10.1145/3582689.
  6. David Arthur and Sergei Vassilvitskii. k-means++: the advantages of careful seeding. In Nikhil Bansal, Kirk Pruhs, and Clifford Stein, editors, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 1027-1035. SIAM, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283494.
  7. Per Austrin, Subhash Khot, and Muli Safra. Inapproximability of vertex cover and independent set in bounded degree graphs. Theory Comput., 7(1):27-43, 2011. URL: https://doi.org/10.4086/TOC.2011.V007A003.
  8. Yossi Azar, Debmalya Panigrahi, and Noam Touitou. Online graph algorithms with predictions. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 35-66, 2022. URL: https://doi.org/10.1137/1.9781611977073.3.
  9. Γ‰tienne Bamas, Andreas Maggiori, and Ola Svensson. The primal-dual method for learning augmented algorithms. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems, NeurIPS, 2020. Google Scholar
  10. Sandip Banerjee, Rafail Ostrovsky, and Yuval Rabani. Min-sum clustering (with outliers). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM, pages 16:1-16:16, 2021. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2021.16.
  11. Yair Bartal, Moses Charikar, and Danny Raz. Approximating min-sum k-clustering in metric spaces. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, pages 11-20, 2001. URL: https://doi.org/10.1145/380752.380754.
  12. Babak Behsaz, Zachary Friggstad, Mohammad R. Salavatipour, and Rohit Sivakumar. Approximation algorithms for min-sum k-clustering and balanced k-median. Algorithmica, 81(3):1006-1030, 2019. URL: https://doi.org/10.1007/S00453-018-0454-1.
  13. Justin Y. Chen, Talya Eden, Piotr Indyk, Honghao Lin, Shyam Narayanan, Ronitt Rubinfeld, Sandeep Silwal, Tal Wagner, David P. Woodruff, and Michael Zhang. Triangle and four cycle counting with predictions in graph streams. In The Tenth International Conference on Learning Representations, ICLR, 2022. Google Scholar
  14. Justin Y. Chen, Piotr Indyk, and Tal Wagner. Streaming algorithms for support-aware histograms. In International Conference on Machine Learning, ICML, pages 3184-3203, 2022. URL: https://proceedings.mlr.press/v162/chen22g.html.
  15. Justin Y. Chen, Sandeep Silwal, Ali Vakilian, and Fred Zhang. Faster fundamental graph algorithms via learned predictions. In International Conference on Machine Learning, ICML, pages 3583-3602, 2022. URL: https://proceedings.mlr.press/v162/chen22v.html.
  16. Michael B. Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. J. ACM, 68(1):3:1-3:39, 2021. URL: https://doi.org/10.1145/3424305.
  17. Vincent Cohen-Addad and Karthik C. S. Inapproximability of clustering in lp metrics. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 519-539, 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 Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2635-2648, 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 Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1493-1530, 2022. URL: https://doi.org/10.1137/1.9781611977073.63.
  20. Vincent Cohen-Addad, Kasper Green Larsen, David Saulpic, Chris Schwiegelshohn, and Omar Ali Sheikh-Omar. Improved coresets for euclidean k-means. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems, NeurIPS, 2022. Google Scholar
  21. Vincent Cohen-Addad, David Saulpic, and Chris Schwiegelshohn. A new coreset framework for clustering. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 169-182, 2021. URL: https://doi.org/10.1145/3406325.3451022.
  22. Artur Czumaj and Christian Sohler. Sublinear-time approximation algorithms for clustering via random sampling. Random Struct. Algorithms, 30(1-2):226-256, 2007. URL: https://doi.org/10.1002/RSA.20157.
  23. Sami Davies, Benjamin Moseley, Sergei Vassilvitskii, and Yuyan Wang. Predictive flows for faster ford-fulkerson. In International Conference on Machine Learning, ICML, volume 202, pages 7231-7248, 2023. URL: https://proceedings.mlr.press/v202/davies23b.html.
  24. Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon, and Yuval Rabani. Approximation schemes for clustering problems. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages 50-58, 2003. URL: https://doi.org/10.1145/780542.780550.
  25. Wenceslas Fernandez de la Vega and Claire Kenyon. A randomized approximation scheme for metric MAX-CUT. J. Comput. Syst. Sci., 63(4):531-541, 2001. URL: https://doi.org/10.1006/JCSS.2001.1772.
  26. Michael Dinitz, Sungjin Im, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Faster matchings via learned duals. In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems, NeurIPS, pages 10393-10406, 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/5616060fb8ae85d93f334e7267307664-Abstract.html.
  27. Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput., 34(5):1129-1146, 2005. URL: https://doi.org/10.1137/S0097539704443057.
  28. Jon C. Ergun, Zhili Feng, Sandeep Silwal, David P. Woodruff, and Samson Zhou. Learning-augmented k-means clustering. In The Tenth International Conference on Learning Representations, ICLR, 2022. Google Scholar
  29. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  30. Sreenivas Gollapudi and Debmalya Panigrahi. Online algorithms for rent-or-buy with expert advice. In Proceedings of the 36th International Conference on Machine Learning, ICML, pages 2319-2327, 2019. URL: http://proceedings.mlr.press/v97/gollapudi19a.html.
  31. Elena Grigorescu, Young-San Lin, Sandeep Silwal, Maoyuan Song, and Samson Zhou. Learning-augmented algorithms for online linear and semidefinite programming. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems, NeurIPS, 2022. Google Scholar
  32. Venkatesan Guruswami and Piotr Indyk. Embeddings and non-approximability of geometric problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 537-538, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644198.
  33. Nili Guttmann-Beck and Refael Hassin. Approximation algorithms for min-sum p-clustering. Discret. Appl. Math., 89(1-3):125-142, 1998. URL: https://doi.org/10.1016/S0166-218X(98)00100-0.
  34. Chen-Yu Hsu, Piotr Indyk, Dina Katabi, and Ali Vakilian. Learning-based frequency estimation algorithms. In 7th International Conference on Learning Representations, ICLR, 2019. Google Scholar
  35. Sungjin Im, Ravi Kumar, Mahshid Montazer Qaem, and Manish Purohit. Online knapsack with frequency predictions. In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems, NeurIPS, pages 2733-2743, 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/161c5c5ad51fcc884157890511b3c8b0-Abstract.html.
  36. 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, pages 332-339, 1994. URL: https://doi.org/10.1145/177424.178042.
  37. Piotr Indyk. A sublinear time approximation scheme for clustering in metric spaces. In 40th Annual Symposium on Foundations of Computer Science, FOCS, pages 154-159, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814587.
  38. Piotr Indyk, Ali Vakilian, and Yang Yuan. Learning-based low-rank approximations. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS, pages 7400-7410, 2019. URL: https://proceedings.neurips.cc/paper/2019/hash/1625abb8e458a79765c62009235e9d5b-Abstract.html.
  39. Anil K Jain, M Narasimha Murty, and Patrick J Flynn. Data clustering: a review. ACM computing surveys (CSUR), 31(3):264-323, 1999. URL: https://doi.org/10.1145/331499.331504.
  40. Arun Jambulapati, Yang P. Liu, and Aaron Sidford. Chaining, group leverage score overestimates, and fast spectral hypergraph sparsification. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 196-206. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585136.
  41. Shaofeng H.-C. Jiang, Erzhi Liu, You Lyu, Zhihao Gavin Tang, and Yubo Zhang. Online facility location with predictions. In The Tenth International Conference on Learning Representations, ICLR, 2022. Google Scholar
  42. Shunhua Jiang, Zhao Song, Omri Weinstein, and Hengjie Zhang. A faster algorithm for solving general lps. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 823-832, 2021. URL: https://doi.org/10.1145/3406325.3451058.
  43. Tanqiu Jiang, Yi Li, Honghao Lin, Yisong Ruan, and David P. Woodruff. Learning-augmented data stream algorithms. In 8th International Conference on Learning Representations, ICLR, 2020. Google Scholar
  44. Narendra Karmarkar. A new polynomial-time algorithm for linear programming. Comb., 4(4):373-396, 1984. URL: https://doi.org/10.1007/BF02579150.
  45. Misha Khodak, Maria-Florina Balcan, Ameet Talwalkar, and Sergei Vassilvitskii. Learning predictions for algorithms with predictions. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems, NeurIPS, 2022. Google Scholar
  46. Subhash Khot. Hardness results for coloring 3 -colorable 3 -uniform hypergraphs. In 43rd Symposium on Foundations of Computer Science (FOCS), Proceedings, pages 23-32, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181879.
  47. Jon M. Kleinberg. An impossibility theorem for clustering. In Advances in Neural Information Processing Systems 15 [Neural Information Processing Systems, NIPS, pages 446-453, 2002. URL: https://proceedings.neurips.cc/paper/2002/hash/43e4e6a6f341e00671e123714de019a8-Abstract.html.
  48. Tim Kraska, Alex Beutel, Ed H. Chi, Jeffrey Dean, and Neoklis Polyzotis. The case for learned index structures. In Proceedings of the 2018 International Conference on Management of Data, SIGMOD Conference, pages 489-504, 2018. URL: https://doi.org/10.1145/3183713.3196909.
  49. Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online scheduling via learned weights. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1859-1877, 2020. URL: https://doi.org/10.1137/1.9781611975994.114.
  50. 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.
  51. James R. Lee. Spectral hypergraph sparsification via chaining. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 207-218. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585165.
  52. Yin Tat Lee and Aaron Sidford. Efficient inverse maintenance and faster algorithms for linear programming. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS, pages 230-249, 2015. URL: https://doi.org/10.1109/FOCS.2015.23.
  53. Yin Tat Lee, Zhao Song, and Qiuyi Zhang. Solving empirical risk minimization in the current matrix multiplication time. In Conference on Learning Theory, COLT, pages 2140-2157, 2019. URL: http://proceedings.mlr.press/v99/lee19a.html.
  54. Yi Li, Honghao Lin, Simin Liu, Ali Vakilian, and David P. Woodruff. Learning the positions in countsketch. In The Eleventh International Conference on Learning Representations, ICLR, 2023. Google Scholar
  55. Honghao Lin, Tian Luo, and David P. Woodruff. Learning augmented binary search trees. In International Conference on Machine Learning, ICML, pages 13431-13440, 2022. URL: https://proceedings.mlr.press/v162/lin22f.html.
  56. Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. J. ACM, 68(4):24:1-24:25, 2021. URL: https://doi.org/10.1145/3447579.
  57. 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.
  58. JirΓ­ Matousek. On approximate geometric k-clustering. Discret. Comput. Geom., 24(1):61-84, 2000. URL: https://doi.org/10.1007/S004540010019.
  59. Michael Mitzenmacher. A model for learned bloom filters and optimizing by sandwiching. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems, NeurIPS, pages 462-471, 2018. URL: https://proceedings.neurips.cc/paper/2018/hash/0f49c89d1e7298bb9930789c8ed59d48-Abstract.html.
  60. Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. In Tim Roughgarden, editor, Beyond the Worst-Case Analysis of Algorithms, pages 646-662. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108637435.037.
  61. Thy Dinh Nguyen, Anamay Chaturvedi, and Huy L. Nguyen. Improved learning-augmented algorithms for k-means and k-medians clustering. In The Eleventh International Conference on Learning Representations, ICLR, 2023. Google Scholar
  62. Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ML predictions. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS, pages 9684-9693, 2018. URL: https://proceedings.neurips.cc/paper/2018/hash/73a427badebe0e32caa2e1fc7530b7f3-Abstract.html.
  63. Leonard J. Schulman. Clustering for edge-cost minimization (extended abstract). In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 547-555, 2000. URL: https://doi.org/10.1145/335305.335373.
  64. Yongho Shin, Changyeol Lee, Gukryeol Lee, and Hyung-Chan An. Improved learning-augmented algorithms for the multi-option ski rental problem via best-possible competitive analysis. In International Conference on Machine Learning, ICML, pages 31539-31561, 2023. URL: https://proceedings.mlr.press/v202/shin23c.html.
  65. Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian J. Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In 2nd International Conference on Learning Representations, ICLR, Conference Track Proceedings, 2014. Google Scholar
  66. Pravin M. Vaidya. Speeding-up linear programming using fast matrix multiplication. In 30th Annual Symposium on Foundations of Computer Science, pages 332-337, 1989. Google Scholar
  67. Pravin M. Vaidya. An algorithm for linear programming which requires o(((m+n)n^2 + (m+n)^1.5n)l) arithmetic operations. Math. Program., 47:175-201, 1990. URL: https://doi.org/10.1007/BF01580859.
  68. Shufan Wang, Jian Li, and Shiqiang Wang. Online algorithms for multi-shop ski rental with machine learned advice. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems, NeurIPS, 2020. Google Scholar
  69. Alexander Wei and Fred Zhang. Optimal robustness-consistency trade-offs for learning-augmented online algorithms. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems, NeurIPS, 2020. Google Scholar
  70. Rui Xu and Donald Wunsch. Survey of clustering algorithms. IEEE Transactions on neural networks, 16(3):645-678, 2005. URL: https://doi.org/10.1109/TNN.2005.845141.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact