Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time

Authors Hjalmar Schulz, André Nichterlein , Rolf Niedermeier , Christopher Weyand

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Author Details

Hjalmar Schulz
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
André Nichterlein
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Rolf Niedermeier
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Christopher Weyand
  • Karlsruhe Institute of Technology, Germany


In memory of Rolf Niedermeier, our colleague, friend, and mentor, who sadly passed away before this paper was published.

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Hjalmar Schulz, André Nichterlein, Rolf Niedermeier, and Christopher Weyand. Applying a Cut-Based Data Reduction Rule for Weighted Cluster Editing in Polynomial Time. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Given an undirected graph, the task in Cluster Editing is to insert and delete a minimum number of edges to obtain a cluster graph, that is, a disjoint union of cliques. In the weighted variant each vertex pair comes with a weight and the edge modifications have to be of minimum overall weight. In this work, we provide the first polynomial-time algorithm to apply the following data reduction rule of Böcker et al. [Algorithmica, 2011] for Weighted Cluster Editing: For a graph G = (V,E), merge a vertex set S ⊆ V into a single vertex if the minimum cut of G[S] is at least the combined cost of inserting all missing edges within G[S] plus the cost of cutting all edges from S to the rest of the graph. Complementing our theoretical findings, we experimentally demonstrate the effectiveness of the data reduction rule, shrinking real-world test instances from the PACE Challenge 2021 by around 24% while previous heuristic implementations of the data reduction rule only achieve 8%.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Correlation Clustering
  • Minimum Cut
  • Maximum s-t-Flow


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  1. Faisal N. Abu-Khzam. On the complexity of multi-parameterized cluster editing. Journal of Discrete Algorithms, 45:26-34, 2017. Google Scholar
  2. Faisal N. Abu-Khzam, Judith Egan, Serge Gaspers, Alexis Shaw, and Peter Shaw. Cluster editing with vertex splitting. In Proceedings of the 5th International Symposium on Combinatorial Optimization (ISCO '18), volume 10856 of LNCS, pages 1-13. Springer, 2018. Google Scholar
  3. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56:89-113, 2004. URL:
  4. Amir Ben-Dor, Ron Shamir, and Zohar Yakhini. Clustering gene expression patterns. Journal of Computational Biology, 6(3-4):281-297, 1999. URL:
  5. Thomas Bläsius, Philipp Fischbeck, Lars Gottesbüren, Michael Hamann, Tobias Heuer, Jonas Spinner, Christopher Weyand, and Marcus Wilhelm. A branch-and-bound algorithm for cluster editing. In Proceedings of the 20th International Symposium on Experimental Algorithms (SEA '22), LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  6. Sebastian Böcker. A golden ratio parameterized algorithm for cluster editing. Journal of Discrete Algorithms, 16:79-89, 2012. URL:
  7. Sebastian Böcker and Jan Baumbach. Cluster Editing. In Proceedings of the 9th Conference on Computability in Europe, CiE 2013, volume 7921 of LNCS, pages 33-44. Springer, 2013. Google Scholar
  8. Sebastian Böcker, Sebastian Briesemeister, and Gunnar W. Klau. Exact algorithms for cluster editing: Evaluation and experiments. Algorithmica, 60(2):316-334, 2011. URL:
  9. Yixin Cao and Jianer Chen. Cluster Editing: Kernelization based on edge cuts. Algorithmica, 64(1):152-169, 2012. Google Scholar
  10. Jianer Chen and Jie Meng. A 2k kernel for the cluster editing problem. Journal of Computer and System Sciences, 78(1):211-220, 2012. Google Scholar
  11. Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Cluster editing in multi-layer and temporal graphs. In Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC '18), volume 123 of LIPIcs, pages 24:1-24:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  12. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. CoRR, abs/2203.00671, 2022. URL:
  13. Michael R. Fellows, Michael A. Langston, Frances A. Rosamond, and Peter Shaw. Efficient parameterized preprocessing for Cluster Editing. In Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT '07), volume 4639 of LNCS, pages 312-321. Springer, 2007. URL:
  14. Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michał Pilipczuk, and Yngve Villanger. Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters. Journal of Computer and System Sciences, 80(7):1430-1447, 2014. Google Scholar
  15. Jens Gramm, Jiong Guo, Falk Hüffner, and Rolf Niedermeier. Graph-modeled data clustering: Exact algorithms for clique generation. Theory of Computing Systems, 38(4):373-392, 2005. Google Scholar
  16. Jiong Guo. A more effective linear kernelization for cluster editing. Theoretical Computer Science, 410(8-10):718-726, 2009. URL:
  17. Sepp Hartung and Holger H. Hoos. Programming by optimisation meets parameterised algorithmics: a case study for cluster editing. In Proceedings of the 9th International Conference on Learning and Intelligent Optimization, LION 2015, volume 8994 of LNCS, pages 43-58. Springer, 2015. URL:
  18. Monika Henzinger, Alexander Noe, and Christian Schulz. Practical fully dynamic minimum cut algorithms. In Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX 2022), pages 13-26. SIAM, 2022. URL:
  19. Monika Henzinger, Alexander Noe, Christian Schulz, and Darren Strash. Finding all global minimum cuts in practice. In Proceedings of the 28th Annual European Symposium on Algorithms (ESA '20), volume 173, pages 59:1-59:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  20. David R Karger. Minimum cuts in near-linear time. Journal of the ACM, 47(1):46-76, 2000. Google Scholar
  21. Leon Kellerhals, Tomohiro Koana, André Nichterlein, and Philipp Zschoche. The PACE 2021 Parameterized Algorithms and Computational Experiments Challenge: Cluster Editing. In Proceedings of the International Symposium on Parameterized and Exact Computation (IPEC '21), volume 214, pages 26:1-26:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  22. Christian Komusiewicz and Johannes Uhlmann. Cluster editing with locally bounded modifications. Discrete Applied Mathematics, 160(15):2259-2270, 2012. Google Scholar
  23. Jason Li. Deterministic mincut in almost-linear time. CoRR, abs/2106.05513, 2021. Google Scholar
  24. Hiroshi Nagamochi, Yoshitaka Nakao, and Toshihide Ibaraki. A fast algorithm for cactus representations of minimum cuts. Japan Journal of Industrial and Applied Mathematics, 17(2):245-264, 2000. Google Scholar
  25. Jean-Claude Picard and Maurice Queyranne. On the structure of all minimum cuts in a network and applications. In Combinatorial Optimization II, pages 8-16. Springer, 1980. Google Scholar
  26. Ron Shamir, Roded Sharan, and Dekel Tsur. Cluster graph modification problems. Discrete Applied Mathematics, 144(1-2):173-182, 2004. Google Scholar
  27. Esther Ulitzsch, Qiwei He, Vincent Ulitzsch, Hendrik Molter, André Nichterlein, Rolf Niedermeier, and Steffi Pohl. Combining clickstream analyses and graph-modeled data clustering for identifying common response processes. Psychometrika, 86(1):190-214, 2021. URL:
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