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Incremental (k, z)-Clustering on Graphs

Authors Emilio Cruciani , Sebastian Forster , Antonis Skarlatos

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Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • (k
  • z)-clustering
  • k-median
  • k-means
  • dynamic graph algorithms

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Abstract

Given a weighted undirected graph, a number of clusters k, and an exponent z, the goal in the (k, z)-clustering problem on graphs is to select k vertices as centers that minimize the sum of the distances raised to the power z of each vertex to its closest center. This problem includes the well-known k-median (z = 1) and k-means (z = 2) clustering problems. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact (k, z)-clustering solution in the induced shortest-path metric.
Prior works by Bhattacharya, Costa, Garg, Lattanzi, and Parotsidis [FOCS 2024] and by Bhattacharya, Costa, and Farokhnejad [STOC 2025] consider the dynamic (k, z)-clustering problem for point sets in metric spaces. These algorithms support adversarial point insertions and deletions under a model with access to pairwise distances. This model differs significantly from the dynamic graph setting, where no oracle access is given to pairwise distances and a single edge update can affect many distances - making these approaches inefficient when applied to graphs. While efficient dynamic k-center approximation algorithms on graphs exist [Cruciani, Forster, Goranci, Nazari, and Skarlatos, SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic (k,z)-clustering problem.
As the main result of this paper, we develop a randomized incremental (k, z)-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of Õ(k m^{1+o(1)} + k^{1+1/(λ)} m), where λ ≥ 1 is an arbitrary fixed constant. Our incremental algorithm also achieves an amortized update time of Õ(k  n^o(1) + k^{1+1/(λ)}) and consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size Õ(k) with a total update time of m^{1+o(1)} (independent of the parameter k) over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant - a property that may be of independent interest.
In the second stage, we maintain a constant-factor approximate (k,z)-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static (k,z)-clustering algorithm.

Cite As Get BibTex

Emilio Cruciani, Sebastian Forster, and Antonis Skarlatos. Incremental (k, z)-Clustering on Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 70:1-70:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.70

Author Details

Emilio Cruciani
  • European University of Rome, Italy
Sebastian Forster
  • Department of Computer Science, University of Salzburg, Austria
Antonis Skarlatos
  • Department of Computer Science, University of Warwick, Coventry, UK

Funding

  • Forster, Sebastian: This work is supported by the Austrian Science Fund (FWF): P 32863-N. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 947702).
  • Skarlatos, Antonis: Antonis Skarlatos is funded by the European Union (ERC grant, DYNALP, 101170133). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

References

  1. Amir Abboud, Vincent Cohen-Addad, Euiwoong Lee, and Pasin Manurangsi. On the fine-grained complexity of approximating k-center in sparse graphs. In Telikepalli Kavitha and Kurt Mehlhorn, editors, 2023 Symposium on Simplicity in Algorithms, SOSA 2023, Florence, Italy, January 23-25, 2023, pages 145-155. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977585.CH14.
  2. Charu C. Aggarwal and Haixun Wang. A Survey of Clustering Algorithms for Graph Data, pages 275-301. Springer US, Boston, MA, 2010. URL: https://doi.org/10.1007/978-1-4419-6045-0_9.
  3. 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):FOCS17-97-FOCS17-156, 2020. URL: https://doi.org/10.1137/18M1171321.
  4. Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein. Algorithms and hardness for diameter in dynamic graphs. 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 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.13.
  5. 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, 33(3):544-562, 2004. URL: https://doi.org/10.1137/S0097539702416402.
  6. Yair Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of 37th Conference on Foundations of Computer Science, pages 184-193. IEEE, 1996. URL: https://doi.org/10.1109/SFCS.1996.548477.
  7. Yair Bartal. On approximating arbitrary metrices by tree metrics. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 161-168, 1998. URL: https://doi.org/10.1145/276698.276725.
  8. Surender Baswana, Sumeet Khurana, and Soumojit Sarkar. Fully dynamic randomized algorithms for graph spanners. ACM Trans. Algorithms, 8(4):35:1-35:51, 2012. URL: https://doi.org/10.1145/2344422.2344425.
  9. 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 Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2677-2727. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH101.
  10. Sayan Bhattacharya, Martín Costa, and Ermiya Farokhnejad. Fully dynamic k-median with near-optimal update time and recourse. In Proceedings of the Symposium on Theory of Computing, STOC 2025, 2024. Google Scholar
  11. Sayan Bhattacharya, Martin Costa, Naveen Garg, Silvio Lattanzi, and Nikos Parotsidis. Fully Dynamic k-Clustering with Fast Update Time and Small Recourse. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 216-227, Los Alamitos, CA, USA, October 2024. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS61266.2024.00023.
  12. Leyla Biabani and Ami Paz. k-center clustering in distributed models. In International Colloquium on Structural Information and Communication Complexity, pages 83-100. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-60603-8_5.
  13. Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median and positive correlation in budgeted optimization. ACM Trans. Algorithms, 13(2), March 2017. URL: https://doi.org/10.1145/2981561.
  14. Jarosław Byrka, Yuhao Guo, Yang Hu, Shi Li, Chengzhang Wan, and Zaixuan Wang. k-clustering via iterative randomized rounding, 2026. URL: https://doi.org/10.48550/arXiv.2604.06046.
  15. Moses Charikar, Chandra Chekuri, Ashish Goel, and Sudipto Guha. Rounding via trees: deterministic approximation algorithms for group steiner trees and k-median. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 114-123, 1998. URL: https://doi.org/10.1145/276698.276719.
  16. Moses Charikar, Sudipto Guha, Éva Tardos, and David B Shmoys. A constant-factor approximation algorithm for the k-median problem. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 1-10, 1999. Google Scholar
  17. Adam Coates and Andrew Y. Ng. Learning feature representations with k-means. In Neural Networks: Tricks of the Trade, pages 561-580. Springer, 2 edition, 2012. URL: https://doi.org/10.1007/978-3-642-35289-8_30.
  18. 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 Symposium on Theory of Computing, STOC 2025, 2025. Google Scholar
  19. Vincent Cohen-Addad, Anupam Gupta, Lunjia Hu, Hoon Oh, and David Saulpic. An improved local search algorithm for k-median. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1556-1612. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.65.
  20. Vincent Cohen-Addad, Niklas Hjuler, Nikos Parotsidis, David Saulpic, and Chris Schwiegelshohn. Fully dynamic consistent facility location. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d'Alché-Buc, Emily B. Fox, and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 3250-3260, 2019. URL: https://proceedings.neurips.cc/paper/2019/hash/fface8385abbf94b4593a0ed53a0c70f-Abstract.html.
  21. Emilio Cruciani, Sebastian Forster, Gramoz Goranci, Yasamin Nazari, and Antonis Skarlatos. Dynamic algorithms for k-center on graphs. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3441-3462. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.123.
  22. Devdatt P Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. Google Scholar
  23. David Eppstein, Sariel Har-Peled, and Anastasios Sidiropoulos. Approximate greedy clustering and distance selection for graph metrics. J. Comput. Geom., 11(1):629-652, 2020. URL: https://doi.org/10.20382/JOCG.V11I1A25.
  24. Hendrik Fichtenberger, Silvio Lattanzi, Ashkan Norouzi-Fard, and Ola Svensson. Consistent k-clustering for general metrics. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2660-2678. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.158.
  25. Sebastian Forster and Antonis Skarlatos. Dynamic consistent k-center clustering with optimal recourse. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 212-254. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.7.
  26. Santo Fortunato. Community detection in graphs. Physics Reports, 486(3):75-174, 2010. URL: https://doi.org/10.1016/j.physrep.2009.11.002.
  27. S Guha, N Mishra, R Motwani, and L O'Callaghan. Clustering data streams. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 359-366. IEEE, 2000. Google Scholar
  28. Anupam Gupta and Kanat Tangwongsan. Simpler analyses of local search algorithms for facility location. arXiv preprint arXiv:0809.2554, 2008. URL: https://arxiv.org/abs/0809.2554.
  29. Pierre Hansen and Brigitte Jaumard. Cluster analysis and mathematical programming. Math. Program., 79:191-215, 1997. URL: https://doi.org/10.1007/BF02614317.
  30. Sariel Har-Peled, Piotr Indyk, and Anastasios Sidiropoulos. Euclidean spanners in high dimensions. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 804-809. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.57.
  31. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. A subquadratic-time algorithm for decremental single-source shortest paths. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1053-1072. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.79.
  32. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. Decremental single-source shortest paths on undirected graphs in near-linear total update time. Journal of the ACM, 65(6):36:1-36:40, 2018. Announced at FOCS 2014. URL: https://doi.org/10.1145/3218657.
  33. Lingxiao Huang and Nisheeth K. Vishnoi. Coresets for clustering in euclidean spaces: importance sampling is nearly optimal. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1416-1429. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384296.
  34. A. K. Jain, M. N. Murty, and P. J. Flynn. Data clustering: A review. ACM Comput. Surv., 31(3):264-323, 1999. URL: https://doi.org/10.1145/331499.331504.
  35. Anil K. Jain. Data clustering: 50 years beyond k-means. Pattern Recognition Letters, 31(8):651-666, 2010. Award winning papers from the 19th International Conference on Pattern Recognition (ICPR). URL: https://doi.org/10.1016/j.patrec.2009.09.011.
  36. Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 731-740, New York, NY, USA, 2002. Association for Computing Machinery. URL: https://doi.org/10.1145/509907.510012.
  37. 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.
  38. Shaofeng H.-C. Jiang, Yaonan Jin, Jianing Lou, and Pinyan Lu. Local search for clustering in almost-linear time. CoRR, abs/2504.03513, 2025. URL: https://doi.org/10.48550/arXiv.2504.03513.
  39. Ce Jin, Yael Kirkpatrick, Virginia Vassilevska Williams, and Nicole Wein. Beyond 2-approximation for k-center in graphs. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 175-211. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.6.
  40. Oded Kariv and S Louis Hakimi. An algorithmic approach to network location problems. i: The p-centers. SIAM journal on applied mathematics, 37(3):513-538, 1979. Google Scholar
  41. Max Dupré la Tour, Monika Henzinger, and David Saulpic. Fully dynamic k-means coreset in near-optimal update time. In 32nd Annual European Symposium on Algorithms, ESA 2024, September 2-4, 2024, Royal Holloway, London, United Kingdom, 2024. Google Scholar
  42. Max Dupré la Tour and David Saulpic. Faster and simpler greedy algorithm for k-median and k-means, 2025. URL: https://arxiv.org/abs/2407.11217.
  43. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 901-910. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488723.
  44. Yang P. Liu. Incremental shortest paths in almost linear time via a modified interior point method. CoRR, abs/2506.19207, 2025. URL: https://doi.org/10.48550/arXiv.2506.19207.
  45. Jakub Łącki and Yasamin Nazari. Near-Optimal Decremental Hopsets with Applications. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 86:1-86:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.86.
  46. Ramgopal R. Mettu and C. Greg Plaxton. The online median problem. SIAM J. Comput., 32(3):816-832, 2003. URL: https://doi.org/10.1137/S0097539701383443.
  47. Ramgopal R Mettu and C Greg Plaxton. Optimal time bounds for approximate clustering. Machine Learning, 56:35-60, 2004. URL: https://doi.org/10.1023/B:MACH.0000033114.18632.E0.
  48. M. E. J Newman. Detecting community structure in networks. The European Physical Journal B, 38, 2004. URL: https://doi.org/10.1140/epjb/e2004-00124-y.
  49. Matthew J. Rattigan, Marc E. Maier, and David D. Jensen. Graph clustering with network structure indices. In Zoubin Ghahramani, editor, Machine Learning, Proceedings of the Twenty-Fourth International Conference (ICML 2007), Corvallis, Oregon, USA, June 20-24, 2007, volume 227 of ACM International Conference Proceeding Series, pages 783-790. ACM, 2007. URL: https://doi.org/10.1145/1273496.1273595.
  50. Satu Elisa Schaeffer. Graph clustering. Comput. Sci. Rev., 1(1):27-64, 2007. URL: https://doi.org/10.1016/J.COSREV.2007.05.001.
  51. Adam Schenker, Mark Last, Horst Bunke, and Abraham Kandel. Comparison of algorithms for web document clustering using graph representations of data. In Ana L. N. Fred, Terry Caelli, Robert P. W. Duin, Aurélio C. Campilho, and Dick de Ridder, editors, Structural, Syntactic, and Statistical Pattern Recognition, Joint IAPR International Workshops, SSPR 2004 and SPR 2004, Lisbon, Portugal, August 18-20, 2004 Proceedings, volume 3138 of Lecture Notes in Computer Science, pages 190-197. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27868-9_19.
  52. Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell., 22(8):888-905, 2000. URL: https://doi.org/10.1109/34.868688.
  53. Arie Tamir. An o(pn²) algorithm for the p-median and related problems on tree graphs. Operations research letters, 19(2):59-64, 1996. URL: https://doi.org/10.1016/0167-6377(96)00021-1.
  54. Mikkel Thorup. Quick k-median, k-center, and facility location for sparse graphs. SIAM Journal on Computing, 34(2):405-432, 2005. URL: https://doi.org/10.1137/S0097539701388884.
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