Document Open Access Logo

Tight Bounds for Clique-Packing Parameterized by Clique-Width

Authors Narek Bojikian , Stefan Kratsch

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.115.pdf
  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized complexity
  • triangle packing
  • clique packing
  • clique-width

Metrics

Abstract

In the d-Clique Packing problem, given a graph G and an integer t, we need to decide whether G contains a set of t pairwise vertex-disjoint cliques of size d each. This generalizes Triangle Packing and it is NP-complete for all d ≥ 3. For each such d, we show how to solve the problem in n^𝒪(k^{d-1}) time where k is the clique-width of the graph (with a k-expression of G given in the input). We complement this by showing that, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem in n^o(k^{d-1}) time for any fixed d ≥ 3, already for the special case of seeking a partition into cliques of size d. Our proof also entails W[1]-hardness of d-Clique Packing (and d-Clique Partitioning) parameterized by clique-width for each d ≥ 3. Our work continues a series of results on ETH-tight bounds for fundamental graph problems started by Fomin et al. (SICOMP 2010+2014) who obtained tight bounds for Max-Cut and Edge Dominating Set.

Cite As Get BibTex

Narek Bojikian and Stefan Kratsch. Tight Bounds for Clique-Packing Parameterized by Clique-Width. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 115:1-115:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.115

Author Details

Narek Bojikian
  • Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany

References

  1. Benjamin Bergougnoux and Mamadou Moustapha Kanté. Fast exact algorithms for some connectivity problems parameterized by clique-width. Theor. Comput. Sci., 782:30-53, 2019. URL: https://doi.org/10.1016/j.tcs.2019.02.030.
  2. Benjamin Bergougnoux, Mamadou Moustapha Kanté, and O-joung Kwon. An optimal XP algorithm for hamiltonian cycle on graphs of bounded clique-width. Algorithmica, 82(6):1654-1674, 2020. URL: https://doi.org/10.1007/S00453-019-00663-9.
  3. Benjamin Bergougnoux, Tuukka Korhonen, and Jesper Nederlof. Tight lower bounds for problems parameterized by rank-width. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 11:1-11:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.11.
  4. Hans L. Bodlaender and Jurriaan Hage. On switching classes, nlc-width, cliquewidth and treewidth. Theor. Comput. Sci., 429:30-35, 2012. URL: https://doi.org/10.1016/J.TCS.2011.12.021.
  5. Narek Bojikian, Vera Chekan, and Stefan Kratsch. Tight bounds for some classical problems parameterized by cutwidth. In Anne Benoit, Haim Kaplan, Sebastian Wild, and Grzegorz Herman, editors, 33rd Annual European Symposium on Algorithms, ESA 2025, Warsaw, Poland, September 15-17, 2025, LIPIcs, pages 13:1-13:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.13.
  6. Narek Bojikian and Stefan Kratsch. A tight monte-carlo algorithm for steiner tree parameterized by clique-width. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 29:1-29:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.29.
  7. Narek Bojikian and Stefan Kratsch. Tight bounds for connected odd cycle transversal parameterized by clique-width. In Akanksha Agrawal and Erik Jan van Leeuwen, editors, 20th International Symposium on Parameterized and Exact Computation, IPEC 2025, Warsaw, Poland, September 17-19, 2025, LIPIcs, pages 19:1-19:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.IPEC.2025.19.
  8. Narek Bojikian and Stefan Kratsch. Tight bounds for clique-packing parameterized by clique-width. arXiv preprint arXiv:2606.31873, 2026. URL: https://doi.org/10.48550/arXiv.2606.31873.
  9. Narek Bojikian and Stefan Kratsch. Tight bounds for feedback vertex set parameterized by clique-width. In Sayan Bhattacharya, Danupon Nanongkai, Michael Benedikt, and Gabriele Puppis, editors, 53rd International Colloquium on Automata, Languages, and Programming, ICALP 2026, Royal Holloway, University of London, Egham, United Kingdom, July 7-10, 2026, volume 374 of LIPIcs, pages 39:1-39:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. URL: https://doi.org/10.4230/LIPIcs.ICALP.2026.39.
  10. Vera Chekan and Stefan Kratsch. Tight algorithmic applications of clique-width generalizations. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, Bordeaux, France, August 28 - September 1, 2023, LIPIcs, pages 35:1-35:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.35.
  11. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  12. 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.
  13. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM J. Comput., 39(5):1941-1956, 2010. URL: https://doi.org/10.1137/080742270.
  14. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM J. Comput., 43(5):1541-1563, 2014. URL: https://doi.org/10.1137/130910932.
  15. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Clique-width III: hamiltonian cycle and the odd case of graph coloring. ACM Trans. Algorithms, 15(1):9:1-9:27, 2019. URL: https://doi.org/10.1145/3280824.
  16. Martin Fürer. A natural generalization of bounded tree-width and bounded clique-width. In Alberto Pardo and Alfredo Viola, editors, LATIN 2014: Theoretical Informatics - 11th Latin American Symposium, Montevideo, Uruguay, March 31 - April 4, 2014. Proceedings, Lecture Notes in Computer Science, pages 72-83. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_7.
  17. Martin Fürer. Multi-clique-width. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, Berkeley, CA, USA, January 9-11, 2017, LIPIcs, pages 14:1-14:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.14.
  18. Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The fine-grained complexity of graph homomorphism parameterized by clique-width. ACM Trans. Algorithms, 20(3):19, 2024. URL: https://doi.org/10.1145/3652514.
  19. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by clique-width. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 59:1-59:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.59.
  20. Yoichi Iwata and Yuichi Yoshida. On the equivalence among problems of bounded width. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 754-765. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_63.
  21. Lars Jaffke, Paloma T. Lima, and Daniel Lokshtanov. b-coloring parameterized by clique-width. Theory Comput. Syst., 68(4):1049-1081, 2024. URL: https://doi.org/10.1007/S00224-023-10132-0.
  22. Ojvind Johansson. Clique-decomposition, nlc-decomposition, and modular decomposition-relationships and results for random graphs. Congressus Numerantium, pages 39-60, 1998. Google Scholar
  23. Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discret. Appl. Math., 126(2-3):197-221, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00198-1.
  24. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM J. Discret. Math., 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
  25. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bull. EATCS, 105:41-72, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/92.
  26. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
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