Essentially Tight Kernels For (Weakly) Closed Graphs

Authors Tomohiro Koana , Christian Komusiewicz , Frank Sommer



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2021.35.pdf
  • Filesize: 0.79 MB
  • 15 pages

Document Identifiers

Author Details

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

Acknowledgements

We would like to thank the anonymous reviewers of ISAAC'21 for their many helpful remarks that have substantially improved the presentation of the results in this paper.

Cite AsGet BibTex

Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Essentially Tight Kernels For (Weakly) Closed Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.35

Abstract

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number γ [Fox et al., SICOMP 2020] in addition to the standard parameter solution size k. The weak closure number γ of a graph is upper-bounded by the minimum of its closure number c and its degeneracy d. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size k^𝒪(γ), k^𝒪(γ), and (γk)^𝒪(γ), respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size k^o(γ) by previous results on their kernelization complexity in degenerate graphs [Cygan et al., ACM TALG 2017]. For Capacitated Vertex Cover, we show that even a kernel of size k^o(c) is unlikely. In contrast, for Connected Vertex Cover, we obtain a problem kernel with 𝒪(ck²) vertices. Moreover, we prove that searching for an induced subgraph of order at least k belonging to a hereditary graph class 𝒢 admits a kernel of size k^𝒪(γ) when 𝒢 contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number c and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Fixed-parameter tractability
  • kernelization
  • c-closure
  • weak γ-closure
  • Independent Set
  • Induced Matching
  • Connected Vertex Cover
  • Ramsey numbers
  • Dominating Set

Metrics

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

References

  1. Piotr Berman, Marek Karpinski, and Alex D. Scott. Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity (ECCC), 049, 2003. Google Scholar
  2. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. Google Scholar
  3. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  4. Marek Cygan, Fabrizio Grandoni, and Danny Hermelin. Tight kernel bounds for problems on graphs with small degeneracy. ACM Transactions on Algorithms, 13(3):43:1-43:22, 2017. Google Scholar
  5. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Kernelization hardness of connectivity problems in d-degenerate graphs. Discrete Applied Mathematics, 160(15):2131-2141, 2012. Google Scholar
  6. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12), pages 68-81. SIAM, 2012. Google Scholar
  7. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and IDs. ACM Transactions on Algorithms, 11(2):13:1-13:20, 2014. Google Scholar
  8. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. 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. Paul Erdös and András Hajnal. Ramsey-type theorems. Discrete Applied Mathematics, 25(1-2):37-52, 1989. Google Scholar
  11. Rok Erman, Łukasz Kowalik, Matjaž Krnc, and Tomasz Waleń. Improved induced matchings in sparse graphs. Discrete Applied Mathematics, 158(18):1994-2003, 2010. Google Scholar
  12. Elaine M. Eschen, Chính T. Hoàng, Jeremy P. Spinrad, and R. Sritharan. On graphs without a C₄ or a diamond. Discrete Applied Mathematics, 159(7):581-587, 2011. Google Scholar
  13. 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
  14. Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981. URL: https://doi.org/10.1007/BF02579457.
  15. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12), pages 104-113. SIAM, 2012. Google Scholar
  16. Edin Husic 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.
  17. Iyad A. Kanj, Michael J. Pelsmajer, Marcus Schaefer, and Ge Xia. On the induced matching problem. Journal of Computer and System Sciences, 77(6):1058-1070, 2011. Google Scholar
  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. In Proceedings of the 31st International Symposium on Algorithms and Computation, (ISAAC '20), volume 181 of LIPIcs, pages 20:1-20:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  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. Stefan Kratsch. Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. ACM Transactions on Algorithms, 10(4):19:1-19:16, 2014. Google Scholar
  22. Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. Google Scholar
  23. Stefan Kratsch, Marcin Pilipczuk, Ashutosh Rai, and Venkatesh Raman. Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs. ACM Transactions on Computation Theory, 7(1):4:1-4:18, 2014. Google Scholar
  24. 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
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