Tight Bounds for Connectivity Problems Parameterized by Cutwidth

Authors Narek Bojikian , Vera Chekan , Falko Hegerfeld , Stefan Kratsch

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Narek Bojikian
  • Humboldt-Universität zu Berlin, Germany
Vera Chekan
  • Humboldt-Universität zu Berlin, Germany
Falko Hegerfeld
  • Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany

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Narek Bojikian, Vera Chekan, Falko Hegerfeld, and Stefan Kratsch. Tight Bounds for Connectivity Problems Parameterized by Cutwidth. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. [Bas A. M. van Geffen et al., 2020] posed this question for Odd Cycle Transversal and Feedback Vertex Set. We answer it for these two and four further problems, namely Connected Vertex Cover, Connected Dominating Set, Steiner Tree, and Connected Odd Cycle Transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al. [Carla Groenland et al., 2022] to solve what we call coloring-like problems. Such problems are defined by a symmetric matrix M over 𝔽₂ indexed by a set of colors. The goal is to count the number (modulo some prime p) of colorings of a graph such that M has a 1-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if M has small rank over 𝔽_p. We apply this result to get our upper bounds for CVC and CDS. The upper bounds for OCT and FVS use a subdivision trick to get below the bounds that matrix rank would yield.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Parameterized complexity
  • connectivity problems
  • cutwidth


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  1. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 67-74. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250801.
  2. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
  3. Narek Bojikian, Vera Chekan, Falko Hegerfeld, and Stefan Kratsch. Tight bounds for connectivity problems parameterized by cutwidth, 2022. URL: https://doi.org/10.48550/arXiv.2212.12385.
  4. Cornelius Brand, Esra Ceylan, Robert Ganian, Christian Hatschka, and Viktoriia Korchemna. Edge-cut width: An algorithmically driven analogue of treewidth based on edge cuts. In Michael A. Bekos and Michael Kaufmann, editors, Graph-Theoretic Concepts in Computer Science - 48th International Workshop, WG 2022, Tübingen, Germany, June 22-24, 2022, Revised Selected Papers, volume 13453 of Lecture Notes in Computer Science, pages 98-113. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_8.
  5. Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A tight lower bound for counting Hamiltonian cycles via matrix rank. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1080-1099. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.70.
  6. 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.
  7. 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. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
  8. 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.
  9. Falko Hegerfeld and Stefan Kratsch. Towards exact structural thresholds for parameterized complexity. In Holger Dell and Jesper Nederlof, editors, 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September 7-9, 2022, Potsdam, Germany, volume 249 of LIPIcs, pages 17:1-17:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.17.
  10. 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.
  11. 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.
  12. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, volume 112 of LIPIcs, pages 47:1-47:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.47.
  13. Nancy G. Kinnersley. The vertex separation number of a graph equals its path-width. Inf. Process. Lett., 42(6):345-350, 1992. URL: https://doi.org/10.1016/0020-0190(92)90234-M.
  14. 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.
  15. 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.
  16. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. URL: https://doi.org/10.1137/16M1104834.
  17. Loïc Magne, Christophe Paul, Abhijat Sharma, and Dimitrios M. Thilikos. Edge-treewidth: Algorithmic and combinatorial properties. CoRR, abs/2112.07524, 2021. URL: http://arxiv.org/abs/2112.07524.
  18. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and gaps: Tight complexity results of general factor problems parameterized by treewidth and cutwidth. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 95:1-95:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.95.
  19. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Comb., 7(1):105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  20. Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. ACM Trans. Comput. Theory, 9(4):18:1-18:36, 2018. URL: https://doi.org/10.1145/3154856.
  21. Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower bounds for dynamic programming on planar graphs of bounded cutwidth. J. Graph Algorithms Appl., 24(3):461-482, 2020. URL: https://doi.org/10.7155/jgaa.00542.
  22. Paul Wollan. The structure of graphs not admitting a fixed immersion. J. Comb. Theory, Ser. B, 110:47-66, 2015. URL: https://doi.org/10.1016/j.jctb.2014.07.003.
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