Document Open Access Logo

Online Correlation Clustering: Simultaneously Optimizing All 𝓁_p-Norms

Authors Sami Davies , Benjamin Moseley , Heather Newman

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.73.pdf
  • Filesize: 1.19 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Online algorithms
Keywords
  • Online algorithms
  • correlation clustering
  • all-norms objective
  • beyond-worst-case analysis

Metrics

Abstract

The 𝓁_p-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the 𝓁₁-norm) and ensuring fairness to individual nodes (the 𝓁_∞-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all 𝓁_p-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously O(log⁴ n)-competitive for all 𝓁_p-norms with high probability, O(log n)-competitive for the 𝓁_∞-norm with high probability, and O(1)-competitive for the 𝓁₁-norm in expectation. This work successfully translates the offline "all-norm" guarantee to the online world.
Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the 𝓁₁-norm is trivially O(1)-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting 𝓁_∞-norm must have a competitive ratio of at least Ξ©(n^{1/3}). This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the 𝓁₁- and 𝓁_∞- norms are nearly tight in the AOS model.

Cite As Get BibTex

Sami Davies, Benjamin Moseley, and Heather Newman. Online Correlation Clustering: Simultaneously Optimizing All 𝓁_p-Norms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 73:1-73:23, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.73

Author Details

Sami Davies
  • Department of EECS, UC Berkeley, CA, USA
Benjamin Moseley
  • Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA
Heather Newman
  • Department of Computer Science, Vassar College, Poughkeepsie, NY, USA

References

  1. Saba Ahmadi, Sainyam Galhotra, Barna Saha, and Roy Schwartz. Fair correlation clustering. arXiv preprint arXiv:2002.03508, 2020. URL: https://arxiv.org/abs/2002.03508.
  2. Sara Ahmadian, Alessandro Epasto, Ravi Kumar, and Mohammad Mahdian. Fair correlation clustering. In International Conference on Artificial Intelligence and Statistics, pages 4195-4205. PMLR, 2020. URL: http://proceedings.mlr.press/v108/ahmadian20a.html.
  3. Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: ranking and clustering. Journal of the ACM, 55(5):1-27, 2008. URL: https://doi.org/10.1145/1411509.1411513.
  4. CJ Argue, Alan Frieze, Anupam Gupta, and Christopher Seiler. Learning from a sample in online algorithms. Advances in Neural Information Processing Systems, 35:13852-13863, 2022. URL: https://dl.acm.org/doi/10.5555/3600270.3601277.
  5. Sepehr Assadi, Aaron Bernstein, and Zachary Langley. Improved bounds for distributed load balancing. arXiv preprint arXiv:2008.04148, 2020. URL: https://arxiv.org/abs/2008.04148.
  6. Yossi Azar, Leah Epstein, Yossi Richter, and Gerhard J Woeginger. All-norm approximation algorithms. Journal of Algorithms, 52(2):120-133, 2004. URL: https://doi.org/10.1016/J.JALGOR.2004.02.003.
  7. Arturs Backurs, Piotr Indyk, Krzysztof Onak, Baruch Schieber, Ali Vakilian, and Tal Wagner. Scalable fair clustering. In International Conference on Machine Learning, pages 405-413. PMLR, 2019. URL: http://proceedings.mlr.press/v97/backurs19a.html.
  8. Eric Balkanski, Jason Chatzitheodorou, and Andreas Maggiori. Cost-free fairness in online correlation clustering. In 36th International Conference on Algorithmic Learning Theory, pages 167-203. PMLR, 2025. URL: https://proceedings.mlr.press/v272/balkanski25b.html.
  9. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56(1):89-113, 2004. URL: https://doi.org/10.1023/B:MACH.0000033116.57574.95.
  10. Suman Bera, Deeparnab Chakrabarty, Nicolas Flores, and Maryam Negahbani. Fair algorithms for clustering. Advances in Neural Information Processing Systems, 32:4955-4966, 2019. URL: https://dl.acm.org/doi/abs/10.5555/3454287.3454733.
  11. Aaron Bernstein, Tsvi Kopelowitz, Seth Pettie, Ely Porat, and Clifford Stein. Simultaneously load balancing for every p-norm, with reassignments. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), pages 51:1-51:14. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.51.
  12. Nairen Cao, Vincent Cohen-Addad, Euiwoong Lee, Shi Li, Alantha Newman, and Lukas Vogl. Understanding the cluster linear program for correlation clustering. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1605-1616, 2024. URL: https://doi.org/10.1145/3618260.3649749.
  13. Nairen Cao, Shi Li, and Jia Ye. Simultaneously Approximating All Norms for Massively Parallel Correlation Clustering. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025), pages 40:1-40:20. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ICALP.2025.40.
  14. Nairen Cao, Steven Roche, and Hsin-Hao Su. Min-Max Correlation Clustering via Neighborhood Similarity. In 33rd Annual European Symposium on Algorithms (ESA 2025), pages 41:1-41:18. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.41.
  15. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing, pages 126-137, 2019. URL: https://doi.org/10.1145/3313276.3316322.
  16. Moses Charikar, Neha Gupta, and Roy Schwartz. Local guarantees in graph cuts and clustering. In International Conference on Integer Programming and Combinatorial Optimization, pages 136-147. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_12.
  17. Shuchi Chawla, Konstantin Makarychev, Tselil Schramm, and Grigory Yaroslavtsev. Near optimal lp rounding algorithm for correlation clustering on complete and complete k-partite graphs. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 219-228, 2015. URL: https://doi.org/10.1145/2746539.2746604.
  18. Flavio Chierichetti, Ravi Kumar, Silvio Lattanzi, and Sergei Vassilvitskii. Fair clustering through fairlets. Advances in Neural Information Processing Systems, 30:5036-5044, 2017. URL: https://dl.acm.org/doi/abs/10.5555/3295222.3295256.
  19. Vincent Cohen-Addad, Silvio Lattanzi, Andreas Maggiori, and Nikos Parotsidis. Online and consistent correlation clustering. In International Conference on Machine Learning, pages 4157-4179. PMLR, 2022. URL: https://proceedings.mlr.press/v162/cohen-addad22a.html.
  20. Vincent Cohen-Addad, Euiwoong Lee, and Alantha Newman. Correlation clustering with sherali-adams. 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 651-661, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00068.
  21. Sami Davies, Benjamin Moseley, and Heather Newman. Fast combinatorial algorithms for min max correlation clustering. In International Conference on Machine Learning, pages 7205-7230. PMLR, 2023. URL: https://proceedings.mlr.press/v202/davies23a.html.
  22. Sami Davies, Benjamin Moseley, and Heather Newman. Simultaneously approximating all lp-norms in correlation clustering. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), pages 52:1-52:20. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.52.
  23. Hendrik Fichtenberger, Silvio Lattanzi, Ashkan Norouzi-Fard, and Ola Svensson. Consistent k-clustering for general metrics. In Proceedings of the 2021 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2660-2678. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.158.
  24. Zachary Friggstad and Ramin Mousavi. Fair correlation clustering with global and local guarantees. In Workshop on Algorithms and Data Structures, pages 414-427. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_30.
  25. Daniel Golovin, Anupam Gupta, Amit Kumar, and Kanat Tangwongsan. All-norms and all-l_p-norms approximation algorithms. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pages 199-210. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2008. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2008.1753.
  26. Xiangyu Guo, Janardhan Kulkarni, Shi Li, and Jiayi Xian. Consistent k-median: Simpler, better and robust. In International Conference on Artificial Intelligence and Statistics, pages 1135-1143. PMLR, 2021. URL: https://proceedings.mlr.press/v130/guo21a.html.
  27. Anupam Gupta, Gregory Kehne, and Roie Levin. Set covering with our eyes wide shut. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4530-4553. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.160.
  28. Swati Gupta, Jai Moondra, and Mohit Singh. Which lp norm is the fairest? approximations for fair facility location across all "p". In Proceedings of the 24th ACM Conference on Economics and Computation, page 817, 2023. URL: https://doi.org/10.1145/3580507.3597664.
  29. Swati Gupta, Jai Moondra, and Mohit Singh. Balancing notions of equity: Trade-offs between fair portfolio sizes and achievable guarantees. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1136-1165. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.33.
  30. Holger SG Heidrich, Jannik Irmai, and Bjoern Andres. A 4-approximation algorithm for min max correlation clustering. In International Conference on Artificial Intelligence and Statistics,, pages 1945-1953. PMLR, 2024. URL: https://proceedings.mlr.press/v238/heidrich24a.html.
  31. Mohammad Reza Karimi Jaghargh, Andreas Krause, Silvio Lattanzi, and Sergei Vassilvtiskii. Consistent online optimization: Convex and submodular. In International Conference on Artificial Intelligence and Statistics, pages 2241-2250. PMLR, 2019. URL: https://proceedings.mlr.press/v89/jaghargh19a.html.
  32. Sanchit Kalhan, Konstantin Makarychev, and Timothy Zhou. Correlation clustering with local objectives. Advances in Neural Information Processing Systems, 32:9346-9355, 2019. URL: https://dl.acm.org/doi/10.5555/3454287.3455125.
  33. Haim Kaplan, David Naori, and Danny Raz. Competitive analysis with a sample and the secretary problem. In Proceedings of the 2020 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2082-2095. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.128.
  34. Haim Kaplan, David Naori, and Danny Raz. Online weighted matching with a sample. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1247-1272. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.52.
  35. Jon Kleinberg, Yuval Rabani, and Γ‰va Tardos. Fairness in routing and load balancing. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 568-578. IEEE, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814631.
  36. Ravi Kumar, Manish Purohit, Aaron Schild, Zoya Svitkina, and Erik Vee. Semi-Online Bipartite Matching. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), volume 124, pages 50:1-50:20. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.50.
  37. Silvio Lattanzi, Benjamin Moseley, Sergei Vassilvitskii, Yuyan Wang, and Rudy Zhou. Robust online correlation clustering. Advances in Neural Information Processing Systems, 34:4688-4698, 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/250dd56814ad7c50971ee4020519c6f5-Abstract.html.
  38. Silvio Lattanzi and Sergei Vassilvitskii. Consistent k-clustering. In International Conference on Machine Learning, pages 1975-1984. PMLR, 2017. URL: http://proceedings.mlr.press/v70/lattanzi17a.html.
  39. Claire Mathieu, Ocan Sankur, and Warren Schudy. Online correlation clustering. In 27th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 573-584. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2010. URL: https://doi.org/10.4230/LIPIcs.STACS.2010.2486.
  40. Gregory J Puleo and Olgica Milenkovic. Correlation clustering with constrained cluster sizes and extended weights bounds. SIAM Journal on Optimization, 25(3):1857-1872, 2015. URL: https://doi.org/10.1137/140994198.
  41. Gregory J. Puleo and Olgica Milenkovic. Correlation clustering and biclustering with locally bounded errors. In International Conference on Machine Learning, pages 869-877. PMLR, 2016. URL: http://proceedings.mlr.press/v48/puleo16.html.
  42. Roy Schwartz and Roded Zats. Fair correlation clustering in general graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), pages 37:1-37:19. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.37.
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