Document Open Access Logo

On Maximum 2-Clubs

Authors Joanne Dumont , Michael Lampis , Mathieu Liedloff , Anthony Perez, Ioan Todinca

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2025.13.pdf
  • Filesize: 0.8 MB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • 2-clubs
  • chordal graphs
  • SETH
  • parameterized algorithms

Metrics

Abstract

We consider the Maximum 2-Club problem where one is given as input an undirected graph G = (V,E) and seeks a subset of vertices S of maximum size such that any pair of vertices in S is connected by a path of length at most 2 in the graph induced by S. This problem is a natural relaxation of the famous Maximum Clique problem where any pair of vertices must be connected by an edge. Maximum 2-Club has been well-studied and is known to be NP-complete even on split graphs. It can be solved exactly in O^*(1.62ⁿ) time, where n denotes the number of vertices of the input graph, while being polynomial-time solvable on several graph classes. Parameterized algorithms for structural parameters have also been considered, leading in particular to an algorithm with a double-exponential dependence in the parameter treewidth. Such an algorithm is actually the best one known for the larger parameter vertex cover size up to a constant in the exponent. We provide new results in both directions. We first prove that the double-exponential dependence for parameter vertex cover size is unavoidable under the Exponential Time Hypothesis (ETH). This answers a question left open by Hartung, Komusiewicz, Nichterlein and Suchỳ [Hartung et al., 2015]. Our result also implies that the problem cannot be solved in time sub-exponential in n even for split graphs. We then provide an exact algorithm for the problem restricted to chordal graphs, running in O^*(1.1996ⁿ) time, by reducing Maximum 2-Club on this class to Maximum Independent Set on arbitrary graphs with the same number of vertices. The same reduction shows that we can enumerate all maximum (and inclusion-wise maximal) 2-clubs of a chordal graph in O^*(3^{n/3}) = O^*(1.4423ⁿ) time. We conclude by providing a construction of split graphs with Ω(3^{n/3}/poly(n)) maximum2-clubs, for some polynomial poly showing that the bound for enumeration is essentially tight.

Cite As Get BibTex

Joanne Dumont, Michael Lampis, Mathieu Liedloff, Anthony Perez, and Ioan Todinca. On Maximum 2-Clubs. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.13

Author Details

Joanne Dumont
  • Université d'Orléans, INSA CVL, LIFO, UR 4022, Orléans, France
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Mathieu Liedloff
  • Université d'Orléans, INSA CVL, LIFO, UR 4022, Orléans, France
Anthony Perez
  • Université d'Orléans, INSA CVL, LIFO, UR 4022, Orléans, France
Ioan Todinca
  • Université d'Orléans, INSA CVL, LIFO, UR 4022, Orléans, France

Acknowledgements

We would like to thank the anonymous referees for providing insightful comments and references which helped improve the presentation of the paper.

References

  1. Yuichi Asahiro, Eiji Miyano, and Kazuaki Samizo. Approximating maximum diameter-bounded subgraphs. In Latin American Symposium on Theoretical Informatics, pages 615-626. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-12200-2_53.
  2. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In International Workshop on Parameterized and Exact Computation, pages 75-85. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-11269-0_6.
  3. Antoine Castillon. Sous-graphes denses: algorithmes et analyse de complexité de certains problèmes de recherche et de couverture. PhD thesis, Université de Lille, 2024. Google Scholar
  4. Maw-Shang Chang, Ling-Ju Hung, Chih-Ren Lin, and Ping-Chen Su. Finding large-clubs in undirected graphs. Computing, 95(9):739-758, 2013. URL: https://doi.org/10.1007/S00607-012-0263-3.
  5. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discret. Math., 86(1-3):165-177, 1990. URL: https://doi.org/10.1016/0012-365X(90)90358-O.
  6. Gabriel Andrew Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71-76, 1961. Google Scholar
  7. Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012. Google Scholar
  8. Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-16533-7.
  9. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  10. Michelle Girvan and Mark E. J. Newman. Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99(12):7821-7826, 2002. URL: https://doi.org/10.1073/pnas.122653799.
  11. Petr A Golovach, Pinar Heggernes, Dieter Kratsch, and Arash Rafiey. Finding clubs in graph classes. Discrete Applied Mathematics, 174:57-65, 2014. URL: https://doi.org/10.1016/J.DAM.2014.04.016.
  12. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. URL: https://doi.org/10.1007/978-3-642-97881-4.
  13. Sepp Hartung, Christian Komusiewicz, and André Nichterlein. Parameterized algorithmics and computational experiments for finding 2-clubs. In International Symposium on Parameterized and Exact Computation, pages 231-241. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-33293-7_22.
  14. Sepp Hartung, Christian Komusiewicz, André Nichterlein, and Ondřej Suchỳ. On structural parameterizations for the 2-club problem. Discrete Applied Mathematics, 185:79-92, 2015. URL: https://doi.org/10.1016/J.DAM.2014.11.026.
  15. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  16. Christian Komusiewicz. Multivariate algorithmics for finding cohesive subnetworks. Algorithms, 9(1):21, 2016. URL: https://doi.org/10.3390/A9010021.
  17. P Sreenivasa Kumar and CE Veni Madhavan. Minimal vertex separators of chordal graphs. Discrete Applied Mathematics, 89(1-3):155-168, 1998. URL: https://doi.org/10.1016/S0166-218X(98)00123-1.
  18. John W. Moon and Leo Moser. On cliques in graphs. Israel Journal of Mathematics, 3(1):23-28, 1965. URL: https://doi.org/10.1007/BF02760024.
  19. Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, and Xuding Zhu. Clustering powers of sparse graphs. Electron. J. Comb., 27(4):4, 2020. URL: https://doi.org/10.37236/9417.
  20. Donald J. Rose, R. Endre Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput., 5:266-283, 1976. URL: https://doi.org/10.1137/0205021.
  21. Bill Rosgen and Lorna Stewart. Complexity results on graphs with few cliques. Discret. Math. Theor. Comput. Sci., 9(1), 2007. URL: https://doi.org/10.46298/DMTCS.387.
  22. Doron Rotem and Jorge Urrutia. Finding maximum cliques in circle graphs. Networks, 11(3):269-278, 1981. URL: https://doi.org/10.1002/NET.3230110305.
  23. Alexander Schäfer. Exact algorithms for s-club finding and related problems. PhD thesis, Friedrich-Schiller-University Jena, 2009. Google Scholar
  24. Alexander Schäfer, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. Parameterized computational complexity of finding small-diameter subgraphs. Optimization Letters, 6:883-891, 2012. URL: https://doi.org/10.1007/S11590-011-0311-5.
  25. David R. Wood. On the maximum number of cliques in a graph. Graphs Comb., 23(3):337-352, 2007. URL: https://doi.org/10.1007/S00373-007-0738-8.
  26. Mingyu Xiao and Hiroshi Nagamochi. Exact algorithms for maximum independent set. Information and Computation, 255:126-146, 2017. URL: https://doi.org/10.1016/J.IC.2017.06.001.
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