Document Open Access Logo

Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman

Authors Michael Levet, Puck Rombach, Nicholas Sieger

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2024.32.pdf
  • Filesize: 0.81 MB
  • 18 pages

Document Identifiers

Author Details

Michael Levet
  • Department of Computer Science, College of Charleston, SC, USA
Puck Rombach
  • Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA
Nicholas Sieger
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

This work was completed in part at the 2022 Graduate Research Workshop in Combinatorics, which was supported in part by NSF grant #1953985 and a generous award from the Combinatorics Foundation.
  • Levet, Michael: Partially supported by J. A. Grochow’s NSF award CISE-2047756 and the University of Colorado Boulder, Department of Computer Science Summer Research Fellowship.

Cite AsGet BibTex

Michael Levet, Puck Rombach, and Nicholas Sieger. Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.32

Abstract

In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in TC². Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width k admits a rank decomposition of width ≤ 2k and height O(log n) (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in TC¹. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by FO + C formulas with 6k+4 variables and quantifier depth O(log n). Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in NC.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Circuit complexity
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph Isomorphism
  • Weisfeiler-Leman
  • Rank-Width
  • Canonization
  • Descriptive Complexity
  • Circuit Complexity

Metrics

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

Related Versions

References

  1. V. Arvind and Piyush P. Kurur. Graph isomorphism is in SPP. Information and Computation, 204(5):835-852, 2006. URL: https://doi.org/10.1016/j.ic.2006.02.002.
  2. L. Babai, E. Luks, and A. Seress. Permutation groups in NC. In STOC 1987, STOC '87, pages 409-420, New York, NY, USA, 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28439.
  3. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In STOC'16 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 684-697. ACM, New York, 2016. Preprint of full version at https://arxiv.org/abs/1512.03547v2. URL: https://doi.org/10.1145/2897518.2897542.
  4. László Babai. Canonical form for graphs in quasipolynomial time: Preliminary report. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 1237-1246, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316356.
  5. Hans L. Bodlaender. NC-algorithms for graphs with small treewidth. In J. van Leeuwen, editor, Graph-Theoretic Concepts in Computer Science, pages 1-10, Berlin, Heidelberg, 1989. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/3-540-50728-0_32.
  6. Harry Buhrman and Steven Homer. Superpolynomial circuits, almost sparse oracles and the exponential hierarchy. In R. K. Shyamasundar, editor, Foundations of Software Technology and Theoretical Computer Science, 12th Conference, New Delhi, India, December 18-20, 1992, Proceedings, volume 652 of Lecture Notes in Computer Science, pages 116-127. Springer, 1992. URL: https://doi.org/10.1007/3-540-56287-7_99.
  7. Jin-Yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12(4):389-410, 1992. Originally appeared in SFCS '89. URL: https://doi.org/10.1007/BF01305232.
  8. Bruno Courcelle and Mamadou Kanté. Graph operations characterizing rank-width and balanced graph expressions. In Graph-Theoretic Concepts in Computer Science, 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007. Revised Papers, pages 66-75, June 2007. URL: https://doi.org/10.1007/978-3-540-74839-7_7.
  9. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  10. Bireswar Das, Anirban Dasgupta, Murali Krishna Enduri, and I. Vinod Reddy. On nc algorithms for problems on bounded rank-width graphs. Information Processing Letters, 139:64-67, 2018. URL: https://doi.org/10.1016/j.ipl.2018.07.007.
  11. Bireswar Das, Jacobo Torán, and Fabian Wagner. Restricted space algorithms for isomorphism on bounded treewidth graphs. Information and Computation, 217:71-83, 2012. URL: https://doi.org/10.1016/j.ic.2012.05.003.
  12. Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. Planar graph isomorphism is in log-space. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 203-214, 2009. URL: https://doi.org/10.1109/CCC.2009.16.
  13. Heinz-Dieter Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer, 2 edition, 1994. URL: https://doi.org/10.1007/978-1-4757-2355-7.
  14. Michael Elberfeld and Ken-ichi Kawarabayashi. Embedding and canonizing graphs of bounded genus in logspace. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 383-392, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591865.
  15. Michael Elberfeld and Pascal Schweitzer. Canonizing graphs of bounded tree width in logspace. ACM Trans. Comput. Theory, 9(3), October 2017. URL: https://doi.org/10.1145/3132720.
  16. Vyacheslav Futorny, Joshua A. Grochow, and Vladimir V. Sergeichuk. Wildness for tensors. Lin. Alg. Appl., 566:212-244, 2019. Preprint arXiv:1810.09219 [math.RT]. URL: https://doi.org/10.1016/j.laa.2018.12.022.
  17. Joshua A. Grochow and Youming Qiao. Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions. arXiv:1907.00309 [cs.CC], 2019. URL: https://doi.org/10.48550/arXiv.1907.00309.
  18. Martin Grohe. Isomorphism testing for embeddable graphs through definability. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC '00, pages 63-72, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335313.
  19. Martin Grohe. Fixed-point definability and polynomial time on graphs with excluded minors. J. ACM, 59(5), November 2012. URL: https://doi.org/10.1145/2371656.2371662.
  20. Martin Grohe and Sandra Kiefer. A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 117:1-117:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.117.
  21. Martin Grohe and Sandra Kiefer. Logarithmic Weisfeiler-Leman Identifies All Planar Graphs. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 134:1-134:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.134.
  22. 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.
  23. Martin Grohe, Daniel Neuen, Pascal Schweitzer, and Daniel Wiebking. An improved isomorphism test for bounded-tree-width graphs. ACM Trans. Algorithms, 16(3), June 2020. URL: https://doi.org/10.1145/3382082.
  24. Martin Grohe and Pascal Schweitzer. Isomorphism testing for graphs of bounded rank width. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1010-1029, 2015. URL: https://doi.org/10.1109/FOCS.2015.66.
  25. Martin Grohe and Oleg Verbitsky. Testing graph isomorphism in parallel by playing a game. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, volume 4051 of Lecture Notes in Computer Science, pages 3-14. Springer, 2006. URL: https://doi.org/10.1007/11786986_2.
  26. Lauri Hella. Definability hierarchies of generalized quantifiers. Annals of Pure and Applied Logic, 43(3):235-271, 1989. URL: https://doi.org/10.1016/0168-0072(89)90070-5.
  27. Lauri Hella. Logical hierarchies in PTIME. Information and Computation, 129(1):1-19, 1996. URL: https://doi.org/10.1006/inco.1996.0070.
  28. Sang il Oum and Paul Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/j.jctb.2005.10.006.
  29. 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.
  30. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  31. Sandra Kiefer, Ilia Ponomarenko, and Pascal Schweitzer. The Weisfeiler-Leman dimension of planar graphs is at most 3. J. ACM, 66(6), November 2019. URL: https://doi.org/10.1145/3333003.
  32. Johannes Köbler, Sebastian Kuhnert, Bastian Laubner, and Oleg Verbitsky. Interval graphs: Canonical representations in logspace. SIAM Journal on Computing, 40(5):1292-1315, 2011. URL: https://doi.org/10.1137/10080395X.
  33. Johannes Köbler, Uwe Schöning, and Jacobo Torán. Graph isomorphism is low for PP. Comput. Complex., 2:301-330, 1992. URL: https://doi.org/10.1007/BF01200427.
  34. Johannes Köbler and Oleg Verbitsky. From invariants to canonization in parallel. In Edward A. Hirsch, Alexander A. Razborov, Alexei Semenov, and Anatol Slissenko, editors, Computer Science - Theory and Applications, pages 216-227, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-79709-8_23.
  35. Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky. On the isomorphism problem for helly circular-arc graphs. Information and Computation, 247:266-277, 2016. URL: https://doi.org/10.1016/j.ic.2016.01.006.
  36. Richard E. Ladner. On the structure of polynomial time reducibility. J. ACM, 22(1):155-171, January 1975. URL: https://doi.org/10.1145/321864.321877.
  37. Leonid Libkin. Elements of Finite Model Theory. Springer, 2004. URL: https://doi.org/10.1007/978-3-662-07003-1_1.
  38. Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. SIAM Journal on Computing, 46(1):161-189, 2017. URL: https://doi.org/10.1137/140999980.
  39. Rudolf Mathon. A note on the graph isomorphism counting problem. Information Processing Letters, 8(3):131-136, 1979. URL: https://doi.org/10.1016/0020-0190(79)90004-8.
  40. Daniel Neuen and Pascal Schweitzer. An exponential lower bound for individualization-refinement algorithms for graph isomorphism. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 138-150. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188900.
  41. Sang-il Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory, 57(3):239-244, 2008. URL: https://doi.org/10.1002/jgt.20280.
  42. Uwe Schöning. Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 37(3):312-323, 1988. URL: https://doi.org/10.1016/0022-0000(88)90010-4.
  43. Thomas Thierauf and Fabian Wagner. The isomorphism problem for planar 3-connected graphs is in unambiguous logspace. Theory Comput. Syst., 47(3):655-673, 2010. URL: https://doi.org/10.1007/S00224-009-9188-4.
  44. Jacobo Torán. On the hardness of graph isomorphism. SIAM J. Comput., 33(5):1093-1108, 2004. URL: https://doi.org/10.1137/S009753970241096X.
  45. Oleg Verbitsky. Planar graphs: Logical complexity and parallel isomorphism tests. In STACS 2007, pages 682-693, Berlin, Heidelberg, 2007. Springer-Verlag. URL: https://doi.org/10.5555/1763424.1763505.
  46. Fabian Wagner. Graphs of bounded treewidth can be canonized in AC¹. In Proceedings of the 6th International Conference on Computer Science: Theory and Applications, CSR'11, pages 209-222, Berlin, Heidelberg, 2011. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-20712-9_16.
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