Document

# Graph Clustering Problems Under the Lens of Parameterized Local Search

## File

LIPIcs.IPEC.2023.20.pdf
• Filesize: 0.83 MB
• 19 pages

## Cite As

Jaroslav Garvardt, Nils Morawietz, André Nichterlein, and Mathias Weller. Graph Clustering Problems Under the Lens of Parameterized Local Search. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.20

## Abstract

Cluster Editing is the problem of finding the minimum number of edge-modifications that transform a given graph G into a cluster graph G', that is, each connected component of G' is a clique. Similarly, in the Cluster Deletion problem, we further restrict the sought cluster graph G' to contain only edges that are also present in G. In this work, we consider the parameterized complexity of a local search variant for both problems: LS Cluster Deletion and LS Cluster Editing. Herein, the input also comprises an integer k and a partition 𝒞 of the vertex set of G that describes an initial cluster graph G^*, and we are to decide whether the "k-move-neighborhood" of G^* contains a cluster graph G' that is "better" (uses less modifications) than G^*. Roughly speaking, two cluster graphs G₁ and G₂ are k-move-neighbors if G₁ can be obtained from G₂ by moving at most k vertices to different connected components. We consider parameterizations by k + 𝓁 for some natural parameters 𝓁, such as the number of clusters in 𝒞, the size of a largest cluster in 𝒞, or the cluster-vertex-deletion number (cvd) of G. Our main lower-bound results are that LS Cluster Editing is W[1]-hard when parameterized by k even if 𝒞 has size two and that both LS Cluster Deletion and LS Cluster Editing are W[1]-hard when parameterized by k + 𝓁, where 𝓁 is the size of the largest cluster of 𝒞. On the positive side, we show that both problems admit an algorithm that runs in k^{𝒪(k)}⋅ cvd^{3k} ⋅ n^{𝒪(1)} time and either finds a better cluster graph or correctly outputs that there is no better cluster graph in the k-move-neighborhood of the initial cluster graph. As an intermediate result, we also obtain an algorithm that solves Cluster Deletion in cvd^{cvd} ⋅ n^{𝒪(1)} time.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• parameterized local search
• permissive local search
• FPT
• W[1]-hardness

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Manuel Aprile, Matthew Drescher, Samuel Fiorini, and Tony Huynh. A tight approximation algorithm for the cluster vertex deletion problem. Math. Program., 197(2):1069-1091, 2023. URL: https://doi.org/10.1007/s10107-021-01744-w.
2. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56:89-113, 2004. URL: https://doi.org/10.1023/B:MACH.0000033116.57574.95.
3. Valentin Bartier, Gabriel Bathie, Nicolas Bousquet, Marc Heinrich, Théo Pierron, and Ulysse Prieto. PACE solver description: μsolver - heuristic track. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 33:1-33:3. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.33.
4. Amir Ben-Dor, Ron Shamir, and Zohar Yakhini. Clustering gene expression patterns. Journal of Computational Biology, 6(3-4):281-297, 1999. URL: https://doi.org/10.1089/106652799318274.
5. Daniel Berend and Tamir Tassa. Improved bounds on bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics, 30(2):185-205, 2010.
6. René van Bevern, Vincent Froese, and Christian Komusiewicz. Parameterizing edge modification problems above lower bounds. Theory of Computing Systems, 62(3):739-770, 2018. URL: https://doi.org/10.1007/s00224-016-9746-5.
7. Thomas Bläsius, Philipp Fischbeck, Lars Gottesbüren, Michael Hamann, Tobias Heuer, Jonas Spinner, Christopher Weyand, and Marcus Wilhelm. PACE solver description: Kapoce: A heuristic cluster editing algorithm. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 31:1-31:4. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.31.
8. Sebastian Böcker. A golden ratio parameterized algorithm for cluster editing. Journal of Discrete Algorithms, 16:79-89, 2012. URL: https://doi.org/10.1016/j.jda.2012.04.005.
9. Édouard Bonnet, Yoichi Iwata, Bart M. P. Jansen, and Lukasz Kowalik. Fine-grained complexity of k-OPT in bounded-degree graphs for solving TSP. In Proceedings of the 27th Annual European Symposium on Algorithms (ESA '19), volume 144 of LIPIcs, pages 23:1-23:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
10. Yixin Cao and Jianer Chen. Cluster Editing: Kernelization based on edge cuts. Algorithmica, 64(1):152-169, 2012.
11. Yixin Cao and Yuping Ke. Improved Kernels for Edge Modification Problems. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:14, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.13.
12. 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 (STOC '15), pages 219-228. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746604.
13. Jianer Chen and Jie Meng. A 2k kernel for the cluster editing problem. Journal of Computer and System Sciences, 78(1):211-220, 2012.
14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
15. Martin Dörnfelder, Jiong Guo, Christian Komusiewicz, and Mathias Weller. On the parameterized complexity of consensus clustering. Theoretical Computer Science, 542:71-82, 2014. URL: https://doi.org/10.1016/j.tcs.2014.05.002.
16. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
17. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, and Yngve Villanger. Local search: Is brute-force avoidable? Journal of Computer and System Sciences, 78(3):707-719, 2012.
18. 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: https://doi.org/10.1007/978-3-540-74240-1_27.
19. Jaroslav Garvardt, Niels Grüttemeier, Christian Komusiewicz, and Nils Morawietz. Parameterized local search for max c-cut. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI 2023, 19th-25th August 2023, Macao, SAR, China, pages 5586-5594. ijcai.org, 2023. URL: https://doi.org/10.24963/ijcai.2023/620.
20. Serge Gaspers, Joachim Gudmundsson, Mitchell Jones, Julián Mestre, and Stefan Rümmele. Turbocharging treewidth heuristics. Algorithmica, 81(2):439-475, 2019. URL: https://doi.org/10.1007/s00453-018-0499-1.
21. Serge Gaspers, Eun Jung Kim, Sebastian Ordyniak, Saket Saurabh, and Stefan Szeider. Don't be strict in local search! In Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence (AAAI '12). AAAI Press, 2012.
22. Martin Josef Geiger. PACE solver description: A simplified threshold accepting approach for the cluster editing problem. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 34:1-34:2. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.34.
23. 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.
24. Niels Grüttemeier and Christian Komusiewicz. On the relation of strong triadic closure and cluster deletion. Algorithmica, 82(4):853-880, 2020. URL: https://doi.org/10.1007/s00453-019-00617-1.
25. Niels Grüttemeier, Christian Komusiewicz, and Nils Morawietz. Efficient Bayesian network structure learning via parameterized local search on topological orderings. In Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI '21), pages 12328-12335. AAAI Press, 2021. Full version available at https://doi.org/10.48550/arXiv.2204.02902. URL: https://ojs.aaai.org/index.php/AAAI/article/view/17463.
26. Jiong Guo. A more effective linear kernelization for cluster editing. Theoretical Computer Science, 410(8-10):718-726, 2009. URL: https://doi.org/10.1016/j.tcs.2008.10.021.
27. Jiong Guo, Sepp Hartung, Rolf Niedermeier, and Ondrej Suchý. The parameterized complexity of local search for TSP, more refined. Algorithmica, 67(1):89-110, 2013.
28. Jiong Guo, Danny Hermelin, and Christian Komusiewicz. Local search for string problems: Brute-force is essentially optimal. Theoretical Computer Science, 525:30-41, 2014.
29. Sepp Hartung and Rolf Niedermeier. Incremental list coloring of graphs, parameterized by conservation. Theoretical Computer Science, 494:86-98, 2013.
30. Giuseppe F. Italiano, Athanasios L. Konstantinidis, and Charis Papadopoulos. Structural parameterization of cluster deletion. In Chun-Cheng Lin, Bertrand M. T. Lin, and Giuseppe Liotta, editors, Proceedings of the 17th International Conference and Workshops on Algorithms and Computation (WALCOM 2023), volume 13973 of Lecture Notes in Computer Science, pages 371-383. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-27051-2_31.
31. Maximilian Katzmann and Christian Komusiewicz. Systematic exploration of larger local search neighborhoods for the minimum vertex cover problem. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI '17), pages 846-852. AAAI Press, 2017.
32. Leon Kellerhals, Tomohiro Koana, André Nichterlein, and Philipp Zschoche. The PACE 2021 parameterized algorithms and computational experiments challenge: Cluster editing. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 26:1-26:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.26.
33. Christian Komusiewicz, Simone Linz, Nils Morawietz, and Jannik Schestag. On the complexity of parameterized local search for the maximum parsimony problem. In Proceedings of the 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023), volume 259 of LIPIcs, pages 18:1-18:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CPM.2023.18.
34. Christian Komusiewicz and Nils Morawietz. Parameterized local search for vertex cover: When only the search radius is crucial. In Proceedings of the 17th International Symposium on Parameterized and Exact Computation (IPEC 2022), volume 249 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.20.
35. Christian Komusiewicz and Frank Sommer. Enumerating connected induced subgraphs: Improved delay and experimental comparison. Discrete Applied Mathematics, 303:262-282, 2021.
36. Christian Komusiewicz and Johannes Uhlmann. Cluster editing with locally bounded modifications. Discrete Applied Mathematics, 160(15):2259-2270, 2012. URL: https://doi.org/10.1016/j.dam.2012.05.019.
37. Shaohua Li, Marcin Pilipczuk, and Manuel Sorge. Cluster editing parameterized above modification-disjoint P₃-packings. In Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS '21), volume 187 of LIPIcs, pages 49:1-49:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.49.
38. Junjie Luo, Hendrik Molter, André Nichterlein, and Rolf Niedermeier. Parameterized dynamic cluster editing. Algorithmica, 83(1):1-44, 2021. URL: https://doi.org/10.1007/s00453-020-00746-y.
39. Dániel Marx. Searching the k-change neighborhood for TSP is W[1]-hard. Operations Research Letters, 36(1):31-36, 2008.
40. Satu Elisa Schaeffer. Graph clustering. Computer Science Review, 1(1):27-64, 2007. URL: https://doi.org/10.1016/j.cosrev.2007.05.001.
41. Ron Shamir, Roded Sharan, and Dekel Tsur. Cluster graph modification problems. Discrete Applied Mathematics, 144(1-2):173-182, 2004. URL: https://doi.org/10.1007/3-540-36379-3_33.
42. Sylwester Swat. PACE solver description: Clues - a heuristic solver for the cluster editing problem. In Proceedings of the 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of LIPIcs, pages 32:1-32:3. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.32.
43. Stefan Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139-145, 2011.
44. Dekel Tsur. Faster parameterized algorithm for cluster vertex deletion. Theory Comput. Syst., 65(2):323-343, 2021. URL: https://doi.org/10.1007/s00224-020-10005-w.
45. Dekel Tsur. Cluster deletion revisited. Information Processing Letters, 173:106171, 2022. URL: https://doi.org/10.1016/j.ipl.2021.106171.
46. 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: https://doi.org/10.1007/s11336-020-09743-0.
X

Feedback for Dagstuhl Publishing