Document Open Access Logo

Max-Distance Sparsification for Diversification and Clustering

Author Soh Kumabe

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ESA.2025.46.pdf
  • Filesize: 0.79 MB
  • 14 pages
HTML5 Logo HTML (experimental)
LIPIcs.ESA.2025.46.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Fixed-Parameter Tractability
  • Diversification
  • Clustering

Metrics

Abstract

Let 𝒟 be a set family that is the solution domain of some combinatorial problem. The max-min diversification problem on 𝒟 is the problem to select k sets from 𝒟 such that the Hamming distance between any two selected sets is at least d. FPT algorithms parameterized by k+𝓁, where 𝓁 = max_{D ∈ 𝒟}|D|, and k+d have been actively studied recently for several specific domains.
This paper provides unified algorithmic frameworks to solve this problem. Specifically, for each parameterization k+𝓁 and k+d, we provide an FPT oracle algorithm for the max-min diversification problem using oracles related to 𝒟. We then demonstrate that our frameworks provide the first FPT algorithms on several new domains 𝒟, including the domain of t-linear matroid intersection, almost 2-SAT, minimum edge s,t-flows, vertex sets of s,t-mincut, vertex sets of edge bipartization, and Steiner trees. We also demonstrate that our frameworks generalize most of the existing domain-specific tractability results.
Our main technical breakthrough is introducing the notion of max-distance sparsifier of 𝒟, a domain on which the max-min diversification problem is equivalent to the same problem on the original domain 𝒟. The core of our framework is to design FPT oracle algorithms that construct a constant-size max-distance sparsifier of 𝒟. Using max-distance sparsifiers, we provide FPT algorithms for the max-min and max-sum diversification problems on 𝒟, as well as k-center and k-sum-of-radii clustering problems on 𝒟, which are also natural problems in the context of diversification and have their own interests.

Cite As Get BibTex

Soh Kumabe. Max-Distance Sparsification for Diversification and Clustering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.46

Author Details

Soh Kumabe
  • CyberAgent, Tokyo, Japan

References

  1. Pankaj Agarwal, Aryan Esmailpour, Xiao Hu, Stavros Sintos, and Jun Yang. Computing a well-representative summary of conjunctive query results. Proceedings of the ACM on Management of Data, 2(5):1-27, 2024. URL: https://doi.org/10.1145/3695835.
  2. Pankaj Agarwal, Sariel Har-Peled, and Kasturi Varadarajan. Approximating extent measures of points. Journal of the ACM (JACM), 51(4):606-635, 2004. URL: https://doi.org/10.1145/1008731.1008736.
  3. Amihood Amir, Jessica Ficler, Liam Roditty, and Oren Sar Shalom. On the efficiency of the hamming c-centerstring problems. In Combinatorial Pattern Matching (CPM), pages 1-10. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-07566-2_1.
  4. Sayan Bandyapadhyay, William Lochet, and Saket Saurabh. FPT constant-approximations for capacitated clustering to minimize the sum of cluster radii. In International Symposium on Computational Geometry (SoCG), volume 258, pages 12:1-12:14, 2023. URL: https://doi.org/10.4230/LIPICS.SOCG.2023.12.
  5. Jørgen Bang-Jensen, Kristine Vitting Klinkby, and Saket Saurabh. k-distinct branchings admits a polynomial kernel. In European Symposium on Algorithms (ESA), pages 11:1-11:15, 2021. URL: https://doi.org/10.4230/LIPICS.ESA.2021.11.
  6. Jørgen Bang-Jensen, Saket Saurabh, and Sven Simonsen. Parameterized algorithms for non-separating trees and branchings in digraphs. Algorithmica, 76:279-296, 2016. URL: https://doi.org/10.1007/S00453-015-0037-3.
  7. Julien Baste, Michael Fellows, Lars Jaffke, Tomáš Masařík, Mateus de Oliveira Oliveira, Geevarghese Philip, and Frances Rosamond. Diversity of solutions: An exploration through the lens of fixed-parameter tractability theory. Artificial Intelligence, 303:103644, 2022. URL: https://doi.org/10.1016/J.ARTINT.2021.103644.
  8. Julien Baste, Lars Jaffke, Tomáš Masařík, Geevarghese Philip, and Günter Rote. FPT algorithms for diverse collections of hitting sets. Algorithms, 12(12):254, 2019. URL: https://doi.org/10.3390/A12120254.
  9. Clément Carbonnel and Emmanuel Hebrard. Propagation via kernelization: The vertex cover constraint. In International Conference on Principles and Practice of Constraint Programming (CP), pages 147-156. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-44953-1_10.
  10. Clément Carbonnel and Emmanuel Hebrard. On the kernelization of global constraints. In International Joint Conference on Artificial Intelligence (IJCAI), pages 578-584, 2017. URL: https://doi.org/10.24963/IJCAI.2017/81.
  11. Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. In ACM Symposium on Theory of Computing (STOC), pages 1-10, 2001. URL: https://doi.org/10.1145/380752.380753.
  12. Xianrun Chen, Dachuan Xu, Yicheng Xu, and Yong Zhang. Parameterized approximation algorithms for sum of radii clustering and variants. In AAAI Conference on Artificial Intelligence, volume 38, pages 20666-20673, 2024. URL: https://doi.org/10.1609/AAAI.V38I18.30053.
  13. Ryan Curtin, Benjamin Moseley, Hung Ngo, XuanLong Nguyen, Dan Olteanu, and Maximilian Schleich. RK-means: Fast clustering for relational data. In International Conference on Artificial Intelligence and Statistics (AISTATS), pages 2742-2752. PMLR, 2020. URL: http://proceedings.mlr.press/v108/curtin20a.html.
  14. Marek Cygan, Fedor Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  15. Erik Demaine, Fedor Fomin, MohammadTaghi Hajiaghayi, and Dimitrios Thilikos. Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs. ACM Transactions on Algorithms (TALG), 1(1):33-47, 2005. URL: https://doi.org/10.1145/1077464.1077468.
  16. Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. On the parameterized complexity of clustering problems for incomplete data. Journal of Computer and System Sciences, 134:1-19, 2023. URL: https://doi.org/10.1016/J.JCSS.2022.12.001.
  17. Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. Determinantal sieving. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-423. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.16.
  18. Paul Erdös and Richard Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 1(1):85-90, 1960. Google Scholar
  19. Aryan Esmailpour and Stavros Sintos. Improved approximation algorithms for relational clustering. Proceedings of the ACM on Management of Data, 2(5):213:1-213:27, 2024. URL: https://doi.org/10.1145/3695831.
  20. Andreas Emil Feldmann and Dániel Marx. The parameterized hardness of the k-center problem in transportation networks. Algorithmica, 82:1989-2005, 2020. URL: https://doi.org/10.1007/S00453-020-00683-W.
  21. Fedor Fomin, Petr Golovach, Lars Jaffke, Geevarghese Philip, and Danil Sagunov. Diverse pairs of matchings. Algorithmica, 86(6):2026-2040, 2024. URL: https://doi.org/10.1007/S00453-024-01214-7.
  22. Fedor Fomin, Petr Golovach, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Diverse collections in matroids and graphs. Mathematical Programming, 204(1):415-447, 2024. URL: https://doi.org/10.1007/S10107-023-01959-Z.
  23. Ryo Funayama, Yasuaki Kobayashi, and Takeaki Uno. Parameterized complexity of finding dissimilar shortest paths. arXiv preprint arXiv:2402.14376, 2024. URL: https://doi.org/10.48550/arXiv.2402.14376.
  24. Tatsuya Gima, Yuni Iwamasa, Yasuaki Kobayashi, Kazuhiro Kurita, Yota Otachi, and Rin Saito. Computing diverse pair of solutions for tractable sat. arXiv preprint arXiv:2412.04016, 2024. URL: https://doi.org/10.48550/arXiv.2412.04016.
  25. Gregory Gutin, Felix Reidl, and Magnus Wahlström. k-distinct in-and out-branchings in digraphs. Journal of Computer and System Sciences, 95:86-97, 2018. URL: https://doi.org/10.1016/J.JCSS.2018.01.003.
  26. Tesshu Hanaka, Yasuaki Kobayashi, Kazuhiro Kurita, and Yota Otachi. Finding diverse trees, paths, and more. In AAAI Conference on Artificial Intelligence, volume 35, pages 3778-3786, 2021. URL: https://doi.org/10.1609/AAAI.V35I5.16495.
  27. Wen-Lian Hsu and George Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1(3):209-215, 1979. URL: https://doi.org/10.1016/0166-218X(79)90044-1.
  28. Tanmay Inamdar and Kasturi Varadarajan. Capacitated sum-of-radii clustering: An FPT approximation. In European Symposium on Algorithms (ESA), 2020. Google Scholar
  29. Neeldhara Misra, Harshil Mittal, and Ashutosh Rai. On the parameterized complexity of diverse SAT. In International Symposium on Algorithms and Computation (ISAAC), volume 322, pages 50:1-50:18, 2024. URL: https://doi.org/10.4230/LIPICS.ISAAC.2024.50.
  30. Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang. Relational algorithms for k-means clustering. In International Colloquium on Automata, Languages, and Programming (ICALP), volume 198, pages 97:1-97:21, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.97.
  31. Yuto Shida, Giulia Punzi, Yasuaki Kobayashi, Takeaki Uno, and Hiroki Arimura. Finding diverse strings and longest common subsequences in a graph. In Symposium on Combinatorial Pattern Matching (CPM), volume 296, pages 27:1-27:19, 2024. URL: https://doi.org/10.4230/LIPICS.CPM.2024.27.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact