Document Open Access Logo

Isomorphism for Tournaments of Small Twin Width

Authors Martin Grohe , Daniel Neuen

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.78.pdf
  • Filesize: 0.78 MB
  • 20 pages

Document Identifiers

Author Details

Martin Grohe
  • RWTH Aachen University, Germany
Daniel Neuen
  • University of Regensburg, Germany

Funding

  • Grohe, Martin: The research is funded by the European Union (ERC, SymSim, 101054974). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We thank the anonymous reviewers of an earlier version of this paper for helpful feedback which, in particular, resulted an improved bound in Theorem 1.2.

Cite AsGet BibTex

Martin Grohe and Daniel Neuen. Isomorphism for Tournaments of Small Twin Width. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.78

Abstract

We prove that isomorphism of tournaments of twin width at most k can be decided in time k^O(log k) n^O(1). This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters. Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is o(n). (Throughout this abstract, n is the order of the input graphs.)

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Finite Model Theory
Keywords
  • tournament isomorphism
  • twin width
  • fixed-parameter tractability
  • Weisfeiler-Leman algorithm

Metrics

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

Related Versions

References

  1. Vikraman Arvind, Ilia N. Ponomarenko, and Grigory Ryabov. Isomorphism testing of k-spanning tournaments is fixed parameter tractable. CoRR, abs/2201.12312, 2022. URL: https://arxiv.org/abs/2201.12312.
  2. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 684-697. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897542.
  3. László Babai and Eugene M. Luks. Canonical labeling of graphs. In David S. Johnson, Ronald Fagin, Michael L. Fredman, David Harel, Richard M. Karp, Nancy A. Lynch, Christos H. Papadimitriou, Ronald L. Rivest, Walter L. Ruzzo, and Joel I. Seiferas, editors, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 171-183. ACM, 1983. URL: https://doi.org/10.1145/800061.808746.
  4. Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearly optimal spectral gap. Comb., 26(5):495-519, 2006. URL: https://doi.org/10.1007/s00493-006-0029-7.
  5. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1977-1996. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.118.
  6. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: max independent set, min dominating set, and coloring. 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 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.35.
  7. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Torunczyk. Twin-width IV: ordered graphs and matrices. 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 924-937. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520037.
  8. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, and Stéphan Thomassé. Twin-width V: linear minors, modular counting, and matrix multiplication. 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 15:1-15:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.15.
  9. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, and Stéphan Thomassé. Twin-width VI: the lens of contraction sequences. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1036-1056. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.45.
  10. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant. Twin-width and polynomial kernels. Algorithmica, 84(11):3300-3337, 2022. URL: https://doi.org/10.1007/s00453-022-00965-5.
  11. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. URL: https://doi.org/10.1145/3486655.
  12. Jin-yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identification. Comb., 12(4):389-410, 1992. URL: https://doi.org/10.1007/BF01305232.
  13. Maria Chudnovsky, Alexandra Ovetsky Fradkin, and Paul D. Seymour. Tournament immersion and cutwidth. J. Comb. Theory, Ser. B, 102(1):93-101, 2012. URL: https://doi.org/10.1016/j.jctb.2011.05.001.
  14. Maria Chudnovsky, Ringi Kim, Chun-Hung Liu, Paul D. Seymour, and Stéphan Thomassé. Domination in tournaments. J. Comb. Theory, Ser. B, 130:98-113, 2018. URL: https://doi.org/10.1016/j.jctb.2017.10.001.
  15. Maria Chudnovsky, Alex Scott, and Paul D. Seymour. Disjoint paths in unions of tournaments. J. Comb. Theory, Ser. B, 135:238-255, 2019. URL: https://doi.org/10.1016/j.jctb.2018.08.007.
  16. Maria Chudnovsky and Paul D. Seymour. A well-quasi-order for tournaments. J. Comb. Theory, Ser. B, 101(1):47-53, 2011. URL: https://doi.org/10.1016/j.jctb.2010.10.003.
  17. John D. Dixon and Brian Mortimer. Permutation Groups, volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996. URL: https://doi.org/10.1007/978-1-4612-0731-3.
  18. Walter Feit and John G. Thompson. Solvability of groups of odd order. Pacific J. Math., 13:775-1029, 1963. Google Scholar
  19. Fedor V. Fomin and Michal Pilipczuk. On width measures and topological problems on semi-complete digraphs. J. Comb. Theory, Ser. B, 138:78-165, 2019. URL: https://doi.org/10.1016/j.jctb.2019.01.006.
  20. Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. First-order interpretations of bounded expansion classes. ACM Trans. Comput. Log., 21(4):29:1-29:41, 2020. URL: https://doi.org/10.1145/3382093.
  21. Jakub Gajarský, Michal Pilipczuk, and Szymon Torunczyk. Stable graphs of bounded twin-width. In Christel Baier and Dana Fisman, editors, LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2 - 5, 2022, pages 39:1-39:12. ACM, 2022. URL: https://doi.org/10.1145/3531130.3533356.
  22. Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider. Weighted model counting with twin-width. In Kuldeep S. Meel and Ofer Strichman, editors, 25th International Conference on Theory and Applications of Satisfiability Testing, SAT 2022, August 2-5, 2022, Haifa, Israel, volume 236 of LIPIcs, pages 15:1-15:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SAT.2022.15.
  23. Colin Geniet and Stéphan Thomassé. First order logic and twin-width in tournaments. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 53:1-53:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.53.
  24. Martin Grohe. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory, volume 47 of Lecture Notes in Logic. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781139028868.
  25. Martin Grohe and Daniel Neuen. Recent advances on the graph isomorphism problem. In Konrad K. Dabrowski, Maximilien Gadouleau, Nicholas Georgiou, Matthew Johnson, George B. Mertzios, and Daniël Paulusma, editors, Surveys in Combinatorics, 2021: Invited lectures from the 28th British Combinatorial Conference, Durham, UK, July 5-9, 2021, pages 187-234. Cambridge University Press, 2021. URL: https://doi.org/10.1017/9781009036214.006.
  26. Martin Grohe and Daniel Neuen. Canonisation and definability for graphs of bounded rank width. ACM Trans. Comput. Log., 24(1):6:1-6:31, 2023. URL: https://doi.org/10.1145/3568025.
  27. Martin Grohe, Daniel Neuen, and Pascal Schweitzer. A faster isomorphism test for graphs of small degree. SIAM J. Comput., 52(6):S18-1, 2023. URL: https://doi.org/10.1137/19m1245293.
  28. Frank Gurski, Dominique Komander, Carolin Rehs, and Sebastian Wiederrecht. Directed width parameters on semicomplete digraphs. In Ding-Zhu Du, Donglei Du, Chenchen Wu, and Dachuan Xu, editors, Combinatorial Optimization and Applications - 15th International Conference, COCOA 2021, Tianjin, China, December 17-19, 2021, Proceedings, volume 13135 of Lecture Notes in Computer Science, pages 615-628. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92681-6_48.
  29. Petr Hlinený and Jan Jedelský. Twin-width of planar graphs is at most 8, and at most 6 when bipartite planar. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 75:1-75:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.75.
  30. Neil Immerman and Eric Lander. Describing graphs: A first-order approach to graph canonization. In Alan L. Selman, editor, Complexity Theory Retrospective: In Honor of Juris Hartmanis on the Occasion of His Sixtieth Birthday, July 5, 1988, pages 59-81. Springer New York, New York, NY, 1990. URL: https://doi.org/10.1007/978-1-4612-4478-3_5.
  31. Thor Johnson, Neil Robertson, Paul D. Seymour, and Robin Thomas. Directed tree-width. J. Comb. Theory, Ser. B, 82(1):138-154, 2001. URL: https://doi.org/10.1006/jctb.2000.2031.
  32. Sandra Kiefer. The Weisfeiler-Leman algorithm: an exploration of its power. ACM SIGLOG News, 7(3):5-27, 2020. URL: https://doi.org/10.1145/3436980.3436982.
  33. Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci., 25(1):42-65, 1982. URL: https://doi.org/10.1016/0022-0000(82)90009-5.
  34. Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. of Math. (2), 182(1):307-325, 2015. URL: https://doi.org/10.4007/annals.2015.182.1.7.
  35. Gary L. Miller. Isomorphism of graphs which are pairwise k-separable. Inf. Control., 56(1/2):21-33, 1983. URL: https://doi.org/10.1016/S0019-9958(83)80048-5.
  36. Jaroslav Nesetril and Patrice Ossona de Mendez. Structural sparsity. Russian Math. Surveys, 71(1):79-107, 2016. URL: https://doi.org/10.4213/rm9688.
  37. Jaroslav Nesetril, Patrice Ossona de Mendez, and Sebastian Siebertz. Structural properties of the first-order transduction quasiorder. In Florin Manea and Alex Simpson, editors, 30th EACSL Annual Conference on Computer Science Logic, CSL 2022, February 14-19, 2022, Göttingen, Germany (Virtual Conference), volume 216 of LIPIcs, pages 31:1-31:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CSL.2022.31.
  38. Daniel Neuen. Hypergraph isomorphism for groups with restricted composition factors. ACM Trans. Algorithms, 18(3):27:1-27:50, 2022. URL: https://doi.org/10.1145/3527667.
  39. Michal Pilipczuk. Tournaments and Optimality: New Results in Parameterized Complexity. PhD thesis, University of Bergen, 2013. Google Scholar
  40. Ilia N. Ponomarenko. Polynomial time algorithms for recognizing and isomorphism testing of cyclic tournaments. Acta Appl. Math., 29(1-2):139-160, 1992. URL: https://doi.org/10.1007/BF00053383.
  41. Joseph J. Rotman. An Introduction to the Theory of Groups, volume 148 of Graduate Texts in Mathematics. Springer-Verlag, New York, fourth edition, 1995. URL: https://doi.org/10.1007/978-1-4612-4176-8.
  42. Pascal Schweitzer. A polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 66:1-66:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.66.
  43. Ákos Seress. Permutation Group Algorithms, volume 152 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2003. URL: https://doi.org/10.1017/CBO9780511546549.
  44. Stéphan Thomassé. A brief tour in twin-width (invited talk). In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 6:1-6:5. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.6.
  45. Boris Weisfeiler. On Construction and Identification of Graphs, volume 558 of Lecture Notes in Mathematics. Springer-Verlag, 1976. Google Scholar
  46. Boris Weisfeiler and Andrei Leman. The reduction of a graph to canonical form and the algebra which appears therein. NTI, Series 2, 1968. English translation by Grigory Ryabov available at URL: https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact