Document Open Access Logo

Tight Lower Bounds for Problems Parameterized by Rank-Width

Authors Benjamin Bergougnoux , Tuukka Korhonen , Jesper Nederlof

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2023.11.pdf
  • Filesize: 1.18 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • rank-width
  • exponential time hypothesis
  • Boolean-width
  • parameterized algorithms
  • independent set
  • dominating set
  • maximum induced matching
  • feedback vertex set

Metrics

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

Abstract

We show that there is no 2^o(k²) n^O(1) time algorithm for Independent Set on n-vertex graphs with rank-width k, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the 2^O(k²) n^O(1) time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known 2^O(k²) n^O(1) time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width k are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for n-vertex graphs.

Cite As Get BibTex

Benjamin Bergougnoux, Tuukka Korhonen, and Jesper Nederlof. Tight Lower Bounds for Problems Parameterized by Rank-Width. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.11

Author Details

Benjamin Bergougnoux
  • University of Warsaw, Poland
Tuukka Korhonen
  • University of Bergen, Norway
Jesper Nederlof
  • Utrecht University, The Netherlands

Funding

  • Korhonen, Tuukka: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Nederlof, Jesper: Supported by the project CRACKNP that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 853234).

Acknowledgements

This work was initiated while the authors attended the “2022 Advances in Parameterized Graph Algorithms” workshop.

References

  1. Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discret. Appl. Math., 23(1):11-24, 1989. URL: https://doi.org/10.1016/0166-218X(89)90031-0.
  2. Rémy Belmonte and Ignasi Sau. On the complexity of finding large odd induced subgraphs and odd colorings. Algorithmica, 83(8):2351-2373, 2021. URL: https://doi.org/10.1007/s00453-021-00830-x.
  3. Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the d-neighbor equivalence: Acyclicity and connectivity constraints. SIAM J. Discret. Math., 35(3):1881-1926, 2021. URL: https://doi.org/10.1137/20M1350571.
  4. 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.
  5. Hajo Broersma, Petr A. Golovach, and Viresh Patel. Tight complexity bounds for FPT subgraph problems parameterized by the clique-width. Theor. Comput. Sci., 485:69-84, 2013. URL: https://doi.org/10.1016/j.tcs.2013.03.008.
  6. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. H-join decomposable graphs and algorithms with runtime single exponential in rankwidth. Discret. Appl. Math., 158(7):809-819, 2010. URL: https://doi.org/10.1016/j.dam.2009.09.009.
  7. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Boolean-width of graphs. Theor. Comput. Sci., 412(39):5187-5204, 2011. URL: https://doi.org/10.1016/j.tcs.2011.05.022.
  8. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theor. Comput. Sci., 511:66-76, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.009.
  9. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci., 46(2):218-270, 1993. URL: https://doi.org/10.1016/0022-0000(93)90004-G.
  10. 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.
  11. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. J. ACM, 65(3):12:1-12:46, 2018. URL: https://doi.org/10.1145/3148227.
  12. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Trans. Algorithms, 18(2):17:1-17:31, 2022. URL: https://doi.org/10.1145/3506707.
  13. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  14. Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width is NP-complete. SIAM J. Discret. Math., 23(2):909-939, 2009. URL: https://doi.org/10.1137/070687256.
  15. 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.
  16. 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.
  17. Fedor V. Fomin and Tuukka Korhonen. Fast FPT-approximation of branchwidth. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 886-899. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519996.
  18. Robert Ganian and Petr Hlinený. On parse trees and myhill-nerode-type tools for handling graphs of bounded rank-width. Discret. Appl. Math., 158(7):851-867, 2010. URL: https://doi.org/10.1016/j.dam.2009.10.018.
  19. Robert Ganian, Petr Hlinený, and Jan Obdrzálek. A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width. Eur. J. Comb., 34(3):680-701, 2013. URL: https://doi.org/10.1016/j.ejc.2012.07.024.
  20. Carla Groenland, Isja Mannens, Jesper Nederlof, and Krisztina Szilágyi. Tight bounds for counting colorings and connected edge sets parameterized by cutwidth. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Marseille, France (Virtual Conference), volume 219 of LIPIcs, pages 36:1-36:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.36.
  21. Petr Hlinený and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012-1032, 2008. URL: https://doi.org/10.1137/070685920.
  22. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  23. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. Theor. Comput. Sci., 795:520-539, 2019. URL: https://doi.org/10.1016/j.tcs.2019.08.006.
  24. Jisu Jeong, Eun Jung Kim, and Sang-il Oum. Finding branch-decompositions of matroids, hypergraphs, and more. SIAM J. Discret. Math., 35(4):2544-2617, 2021. URL: https://doi.org/10.1137/19M1285895.
  25. 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.
  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.
  27. Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Trans. Algorithms, 5(1):10:1-10:20, 2008. URL: https://doi.org/10.1145/1435375.1435385.
  28. Sang-il Oum, Sigve Hortemo Sæther, and Martin Vatshelle. Faster algorithms for vertex partitioning problems parameterized by clique-width. Theor. Comput. Sci., 535:16-24, 2014. URL: https://doi.org/10.1016/j.tcs.2014.03.024.
  29. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. J. Comb. Theory, Ser. B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/j.jctb.2005.10.006.
  30. Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, Norway, 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact