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Deciding Twin-Width at Most 4 Is NP-Complete

Authors Pierre Bergé, Édouard Bonnet , Hugues Déprés

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Author Details

Pierre Bergé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Hugues Déprés
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

Funding

This paper was supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014-01) and Digraphs (ANR-19-CE48-0013-01).

Acknowledgements

We wish to thank Eunjung Kim, Stéphan Thomassé, and Rémi Watrigant.

Cite AsGet BibTex

Pierre Bergé, Édouard Bonnet, and Hugues Déprés. Deciding Twin-Width at Most 4 Is NP-Complete. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.18

Abstract

We show that determining if an n-vertex graph has twin-width at most 4 is NP-complete, and requires time 2^Ω(n/log n) unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that n-vertex graphs subdivided at least 2 log n times have twin-width at most 4. We also show how to encode trigraphs H (2-edge colored graphs involved in the definition of twin-width) into graphs G, in the sense that every d-sequence (sequence of vertex contractions witnessing that the twin-width is at most d) of G inevitably creates H as an induced subtrigraph, whereas there exists a partial d-sequence that actually goes from G to H. We believe that these facts and their proofs can be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • Twin-width
  • lower bounds

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References

  1. Stefan Arnborg, Derek G Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8(2):277-284, 1987. Google Scholar
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  3. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1977-1996, 2021. URL: https://doi.org/10.1137/1.9781611976465.118.
  4. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Toruńczyk. Twin-width IV: ordered graphs and matrices. CoRR, abs/2102.03117, 2021. URL: http://arxiv.org/abs/2102.03117.
  5. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant. Twin-width and polynomial kernels. In Petr A. Golovach and Meirav Zehavi, editors, 16th International Symposium on Parameterized and Exact Computation, IPEC 2021, September 8-10, 2021, Lisbon, Portugal, volume 214 of LIPIcs, pages 10:1-10:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.10.
  6. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 601-612. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00062.
  7. Derek G. Corneil, Michel Habib, Jean-Marc Lanlignel, Bruce A. Reed, and Udi Rotics. Polynomial-time recognition of clique-width ⩽3 graphs. Discret. Appl. Math., 160(6):834-865, 2012. URL: https://doi.org/10.1016/j.dam.2011.03.020.
  8. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight lower bounds on graph embedding problems. J. ACM, 64(3):18:1-18:22, 2017. URL: https://doi.org/10.1145/3051094.
  9. Anuj Dawar and Stephan Kreutzer. Parameterized complexity of first-order logic. Electronic Colloquium on Computational Complexity (ECCC), 16:131, 2009. URL: http://eccc.hpi-web.de/report/2009/131.
  10. Markus Sortland Dregi and Daniel Lokshtanov. Parameterized complexity of bandwidth on trees. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 405-416. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_34.
  11. Zdenek Dvorák, Daniel Král, and Robin Thomas. Testing first-order properties for subclasses of sparse graphs. J. ACM, 60(5):36:1-36:24, 2013. URL: https://doi.org/10.1145/2499483.
  12. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput., 38(2):629-657, 2008. URL: https://doi.org/10.1137/05064299X.
  13. 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.
  14. Jörg Flum and Martin Grohe. Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput., 31(1):113-145, 2001. URL: https://doi.org/10.1137/S0097539799360768.
  15. Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, and Meirav Zehavi. Computation of hadwiger number and related contraction problems: Tight lower bounds. ACM Trans. Comput. Theory, 13(2):10:1-10:25, 2021. URL: https://doi.org/10.1145/3448639.
  16. Jakub Gajarský, Petr Hlinený, Daniel Lokshtanov, Jan Obdrzálek, Sebastian Ordyniak, M. S. Ramanujan, and Saket Saurabh. FO model checking on posets of bounded width. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 963-974, 2015. URL: https://doi.org/10.1109/FOCS.2015.63.
  17. Jakub Gajarský, Petr Hlinený, Jan Obdrzálek, and Sebastian Ordyniak. Faster existential FO model checking on posets. Logical Methods in Computer Science, 11(4), 2015. URL: https://doi.org/10.2168/LMCS-11(4:8)2015.
  18. Robert Ganian, Petr Hlinený, Daniel Král, Jan Obdrzálek, Jarett Schwartz, and Jakub Teska. FO model checking of interval graphs. Logical Methods in Computer Science, 11(4), 2015. URL: https://doi.org/10.2168/LMCS-11(4:11)2015.
  19. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  20. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3):17:1-17:32, 2017. URL: https://doi.org/10.1145/3051095.
  21. Sylvain Guillemot and Dániel Marx. Finding small patterns in permutations in linear time. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 82-101, 2014. URL: https://doi.org/10.1137/1.9781611973402.7.
  22. Michel Habib and Christophe Paul. A simple linear time algorithm for cograph recognition. Discret. Appl. Math., 145(2):183-197, 2005. URL: https://doi.org/10.1016/j.dam.2004.01.011.
  23. 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.
  24. 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.
  25. 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.
  26. Toshinobu Kashiwabara. NP-completeness of the problem of finding a minimal-clique number interval graph containing a given graph as a subgraph. In Proc. 1979 Int. Symp. Circuit Syst, pages 657-660, 1979. Google Scholar
  27. Tuukka Korhonen. Single-exponential time 2-approximation algorithm for treewidth. CoRR, abs/2104.07463, 2021. URL: http://arxiv.org/abs/2104.07463.
  28. Thomas Lengauer. Black-white pebbles and graph separation. Acta Informatica, 16:465-475, 1981. URL: https://doi.org/10.1007/BF00264496.
  29. Tatsuo Ohtsuki, Hajimu Mori, Ernest S. Kuh, Toshinobu Kashiwabara, and Toshio Fujisawa. One-dimensional logic gate assignment and interval graphs. In The IEEE Computer Society’s Third International Computer Software and Applications Conference, COMPSAC 1979, 6-8 November, 1979, Chicago, Illinois, USA, pages 101-106. IEEE, 1979. URL: https://doi.org/10.1109/CMPSAC.1979.762474.
  30. 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.
  31. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  32. Sigve Hortemo Sæther and Martin Vatshelle. Hardness of computing width parameters based on branch decompositions over the vertex set. Theor. Comput. Sci., 615:120-125, 2016. URL: https://doi.org/10.1016/j.tcs.2015.11.039.
  33. James B. Saxe. Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time. SIAM J. Algebraic Discret. Methods, 1(4):363-369, 1980. URL: https://doi.org/10.1137/0601042.
  34. André Schidler and Stefan Szeider. A SAT approach to twin-width. CoRR (accepted at ALENEX '22), abs/2110.06146, 2021. URL: http://arxiv.org/abs/2110.06146.
  35. Craig A. Tovey. A simplified NP-complete satisfiability problem. Discret. Appl. Math., 8(1):85-89, 1984. URL: https://doi.org/10.1016/0166-218X(84)90081-7.
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