Document Open Access Logo

Tight Algorithmic Applications of Clique-Width Generalizations

Authors Vera Chekan , Stefan Kratsch

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2023.35.pdf
  • Filesize: 0.74 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

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

Metrics

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

Abstract

In this work, we study two natural generalizations of clique-width introduced by Martin Fürer. Multi-clique-width (mcw) allows every vertex to hold multiple labels [ITCS 2017], while for fusion-width (fw) we have a possibility to merge all vertices of a certain label [LATIN 2014]. Fürer has shown that both parameters are upper-bounded by treewidth thus making them more appealing from an algorithmic perspective than clique-width and asked for applications of these parameters for problem solving. First, we determine the relation between these two parameters by showing that mcw ≤ fw + 1. Then we show that when parameterized by multi-clique-width, many problems (e.g., Connected Dominating Set) admit algorithms with the same running time as for clique-width despite the exponential gap between these two parameters. For some problems (e.g., Hamiltonian Cycle) we show an analogous result for fusion-width: For this we present an alternative view on fusion-width by introducing so-called glue-expressions which might be interesting on their own. All algorithms obtained in this work are tight up to (Strong) Exponential Time Hypothesis.

Cite As Get BibTex

Vera Chekan and Stefan Kratsch. Tight Algorithmic Applications of Clique-Width Generalizations. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.35

Author Details

Vera Chekan
  • Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany

Funding

  • Chekan, Vera: Supported by DFG Research Training Group 2434 "Facets of Complexity".

References

  1. 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.
  2. 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.
  3. Hans L. Bodlaender and Klaus Jansen. On the complexity of the maximum cut problem. Nord. J. Comput., 7(1):14-31, 2000. Google Scholar
  4. Hans L. Bodlaender, Erik Jan van Leeuwen, Johan M. M. van Rooij, and Martin Vatshelle. Faster algorithms on branch and clique decompositions. In Petr Hlinený and Antonín Kucera, editors, Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings, volume 6281 of Lecture Notes in Computer Science, pages 174-185. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15155-2_17.
  5. 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.
  6. 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.
  7. 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.
  8. Vera Chekan and Stefan Kratsch. Tight algorithmic applications of clique-width generalizations, 2023. Technical report. URL: https://doi.org/10.48550/arXiv.2307.04628.
  9. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM J. Comput., 34(4):825-847, 2005. URL: https://doi.org/10.1137/S0097539701385351.
  10. Bruno Courcelle and Johann A. Makowsky. Fusion in relational structures and the verification of monadic second-order properties. Math. Struct. Comput. Sci., 12(2):203-235, 2002. URL: https://doi.org/10.1017/S0960129501003565.
  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. Eduard Eiben, Robert Ganian, Thekla Hamm, Lars Jaffke, and O-joung Kwon. A unifying framework for characterizing and computing width measures. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 63:1-63:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.63.
  14. Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve np-hard graph problems on clique-width bounded graphs in polynomial time. In Andreas Brandstädt and Van Bang Le, editors, Graph-Theoretic Concepts in Computer Science, 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14-16, 2001, Proceedings, volume 2204 of Lecture Notes in Computer Science, pages 117-128. Springer, 2001. URL: https://doi.org/10.1007/3-540-45477-2_12.
  15. 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.
  16. Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, and Philip Wellnitz. Tight complexity bounds for counting generalized dominating sets in bounded-treewidth graphs. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3664-3683. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch140.
  17. 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.
  18. 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.
  19. 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, volume 8392 of Lecture Notes in Computer Science, pages 72-83. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_7.
  20. Martin Fürer. Multi-clique-width. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 14:1-14:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.14.
  21. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by clique-width. CoRR, abs/2302.03627, 2023. URL: https://doi.org/10.48550/arXiv.2302.03627.
  22. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by modular-treewidth. CoRR, abs/2302.14128, 2023. URL: https://doi.org/10.48550/arXiv.2302.14128.
  23. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  24. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1802-1811. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.130.
  25. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters, tight bounds, and approximation for (k, r)-center. Discret. Appl. Math., 264:90-117, 2019. URL: https://doi.org/10.1016/j.dam.2018.11.002.
  26. 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.
  27. 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.
  28. 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.
  29. Stefan Mengel. Parameterized compilation lower bounds for restricted cnf-formulas. In Nadia Creignou and Daniel Le Berre, editors, Theory and Applications of Satisfiability Testing - SAT 2016 - 19th International Conference, Bordeaux, France, July 5-8, 2016, Proceedings, volume 9710 of Lecture Notes in Computer Science, pages 3-12. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-40970-2_1.
  30. 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.
  31. Daniël Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for CNF formulas of bounded modular treewidth. Algorithmica, 76(1):168-194, 2016. URL: https://doi.org/10.1007/s00453-015-0030-x.
  32. Marcin Pilipczuk. A tight lower bound for vertex planarization on graphs of bounded treewidth. Discret. Appl. Math., 231:211-216, 2017. URL: https://doi.org/10.1016/j.dam.2016.05.019.
  33. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Amos Fiat and Peter Sanders, editors, Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_51.
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