Document Open Access Logo

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Authors Tomohiro Koana , Christian Komusiewicz , Frank Sommer

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2020.20.pdf
  • Filesize: 0.54 MB
  • 17 pages

Document Identifiers

Author Details

Tomohiro Koana
  • Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
Christian Komusiewicz
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
Frank Sommer
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany

Funding

  • Koana, Tomohiro: Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/19.
  • Sommer, Frank: Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.

Cite AsGet BibTex

Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Computing Dense and Sparse Subgraphs of Weakly Closed Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.20

Abstract

A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩ N(v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class 𝒢 which is closed under taking subgraphs admits a kernel with at most γ k² vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Fixed-parameter tractability
  • c-closure
  • degeneracy
  • clique relaxations
  • bicliques
  • dominating set

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Noga Alon and Shai Gutner. Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica, 54(4):544-556, 2009. Google Scholar
  2. Alessio Conte, Donatella Firmani, Caterina Mordente, Maurizio Patrignani, and Riccardo Torlone. Fast enumeration of large k-plexes. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '17), pages 115-124. ACM, 2017. Google Scholar
  3. Alessio Conte, Tiziano De Matteis, Daniele De Sensi, Roberto Grossi, Andrea Marino, and Luca Versari. D2K: scalable community detection in massive networks via small-diameter k-plexes. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '18), pages 1272-1281. ACM, 2018. Google Scholar
  4. Jean-François Couturier and Dieter Kratsch. Bicolored independent sets and bicliques. Information Processing Letters, 112(8-9):329-334, 2012. Google Scholar
  5. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  6. Marek Cygan, Geevarghese Philip, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Dominating set is fixed parameter tractable in claw-free graphs. Theoretical Computer Science, 412(50):6982-7000, 2011. Google Scholar
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  8. David Eppstein. Arboricity and bipartite subgraph listing algorithms. Information Processing Letters, 51(4):207-211, 1994. Google Scholar
  9. David Eppstein, Maarten Löffler, and Darren Strash. Listing all maximal cliques in large sparse real-world graphs. ACM Journal of Experimental Algorithmics, 18, 2013. Google Scholar
  10. Qilong Feng, Shaohua Li, Zeyang Zhou, and Jianxin Wang. Parameterized algorithms for edge biclique and related problems. Theoretical Computer Science, 734:105-118, 2018. Google Scholar
  11. Jacob Fox, Tim Roughgarden, C. Seshadhri, Fan Wei, and Nicole Wein. Finding cliques in social networks: A new distribution-free model. SIAM Journal on Computing, 49(2):448-464, 2020. Google Scholar
  12. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  13. Serge Gaspers, Dieter Kratsch, and Mathieu Liedloff. On independent sets and bicliques in graphs. Algorithmica, 62(3-4):637-658, 2012. Google Scholar
  14. Petr A. Golovach and Yngve Villanger. Parameterized complexity for domination problems on degenerate graphs. In Proceedings of the 34th International Workshop Graph-Theoretic Concepts in Computer Science (WG '08), volume 5344 of Lecture Notes in Computer Science, pages 195-205, 2008. Google Scholar
  15. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. Journal of the ACM, 64(3):17:1-17:32, 2017. Google Scholar
  16. Danny Hermelin and George Manoussakis. Efficient enumeration of maximal induced bicliques. Discrete Applied Mathematics, 2020. To appear. URL: https://doi.org/10.1016/j.dam.2020.04.034.
  17. Edin Husić and Tim Roughgarden. FPT algorithms for finding dense subgraphs in c-closed graphs. CoRR, abs/2007.09768, 2020. URL: http://arxiv.org/abs/2007.09768.
  18. Subhash Khot and Venkatesh Raman. Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289(2):997-1008, 2002. Google Scholar
  19. Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Computing dense and sparse subgraphs of weakly closed graphs. CoRR, abs/2007.05630, 2020. URL: http://arxiv.org/abs/2007.05630.
  20. Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Exploiting c-closure in kernelization algorithms for graph problems. In Proceedings of the 28th Annual European Symposium on Algorithms (ESA '20), volume 173 of LIPIcs, pages 65:1-65:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  21. Tomohiro Koana and André Nichterlein. Detecting and enumerating small induced subgraphs in c-closed graphs. CoRR, abs/2007.12077, 2020. URL: http://arxiv.org/abs/2007.12077.
  22. Christian Komusiewicz. Multivariate algorithmics for finding cohesive subnetworks. Algorithms, 9(1):21, 2016. Google Scholar
  23. Christian Komusiewicz, Falk Hüffner, Hannes Moser, and Rolf Niedermeier. Isolation concepts for efficiently enumerating dense subgraphs. Theoretical Computer Science, 410(38-40):3640-3654, 2009. Google Scholar
  24. Christian Komusiewicz and Manuel Sorge. An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems. Discrete Applied Mathematics, 193:145-161, 2015. Google Scholar
  25. Bingkai Lin. The parameterized complexity of the k-biclique problem. Journal of the ACM, 65(5):34:1-34:23, 2018. Google Scholar
  26. Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25(1):42-65, 1982. Google Scholar
  27. René Peeters. The maximum edge biclique problem is NP-complete. Discrete Applied Mathematics, 131(3):651-654, 2003. Google Scholar
  28. Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond. ACM Transactions on Algorithms, 9(1):11:1-11:23, 2012. Google Scholar
  29. Venkatesh Raman and Saket Saurabh. Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica, 52(2):203-225, 2008. Google Scholar
  30. Detlef Seese. Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science, 6(6):505-526, 1996. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact