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Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Authors Frank Fuhlbrück, Johannes Köbler, Oleg Verbitsky

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LIPIcs.STACS.2020.43.pdf
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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Graph Isomorphism
  • Weisfeiler-Leman Algorithm
  • Cai-Fürer-Immerman Graphs
  • coherent Configurations

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Abstract

It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges.
Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.

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Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.43

Author Details

Frank Fuhlbrück
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Johannes Köbler
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Oleg Verbitsky
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany

Funding

  • Verbitsky, Oleg: Supported by DFG grant KO 1053/8 - 1. On leave from the IAPMM, Lviv, Ukraine.

Acknowledgements

We thank Ilia Ponomarenko and the anonymous referees for their numerous detailed comments and Daniel Neuen and Pascal Schweitzer for a useful discussion of multipede graphs. The third author is especially grateful to Ilia Ponomarenko for his patient and insightful guidance through the theory of coherent configurations.

References

  1. Vikraman Arvind and Johannes Köbler. On hypergraph and graph isomorphism with bounded color classes. In 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS'06), volume 3884 of Lecture Notes in Computer Science, pages 384-395. Springer, 2006. URL: https://doi.org/10.1007/11672142_31.
  2. Vikraman Arvind, Johannes Köbler, Gaurav Rattan, and Oleg Verbitsky. Graph isomorphism, color refinement, and compactness. Computational Complexity, 26(3):627-685, 2017. URL: https://doi.org/10.1007/s00037-016-0147-6.
  3. Vikraman Arvind, Piyush P. Kurur, and T. C. Vijayaraghavan. Bounded color multiplicity graph isomorphism is in the #L hierarchy. In 20th Annual IEEE Conference on Computational Complexity (CCC'05), pages 13-27. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/CCC.2005.7.
  4. László Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC'16), pages 684-697, 2016. URL: https://doi.org/10.1145/2897518.2897542.
  5. Béla Bollobás. Distinguishing vertices of random graphs. Annals of Discrete Mathematics, 13:33-49, 1982. URL: https://doi.org/10.1016/S0304-0208(08)73545-X.
  6. R. C. Bose and Dale M. Mesner. On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist., 30:21-38, 1959. URL: https://doi.org/10.1214/aoms/1177706356.
  7. James R. Bunch and John E. Hopcroft. Triangular factorization and inversion by fast matrix multiplication. Math. Comp., 28:231-236, 1974. URL: https://doi.org/10.2307/2005828.
  8. Gerhard Buntrock, Carsten Damm, Ulrich Hertrampf, and Christoph Meinel. Structure and importance of logspace-mod classes. Mathematical systems theory, 25(3):223-237, 1992. URL: https://doi.org/10.1007/BF01374526.
  9. Jin-yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identifications. Combinatorica, 12(4):389-410, 1992. URL: https://doi.org/10.1007/BF01305232.
  10. Gang Chen and Ilia Ponomarenko. Lectures on coherent configurations. http://www.pdmi.ras.ru/~inp/ccNOTES.pdf, 2019. URL: http://www.pdmi.ras.ru/~inp/ccNOTES.pdf.
  11. Ho Yee Cheung, Tsz Chiu Kwok, and Lap Chi Lau. Fast matrix rank algorithms and applications. J. ACM, 60(5):31:1-31:25, 2013. URL: https://doi.org/10.1145/2528404.
  12. Anuj Dawar and Kashif Khan. Constructing hard examples for graph isomorphism. J. Graph Algorithms Appl., 23(2):293-316, 2019. URL: https://doi.org/10.7155/jgaa.00492.
  13. Bart de Bruyn. An introduction to incidence geometry. Front. Math. Basel: Birkhäuser/Springer, 2016. Google Scholar
  14. S. Evdokimov and I. Ponomarenko. Characterization of cyclotomic schemes and normal schur rings over a cyclic group. St. Petersburg Math. J., 14(2):189-221, 2003. Google Scholar
  15. Sergei Evdokimov and Ilia Ponomarenko. On highly closed cellular algebras and highly closed isomorphisms. Electr. J. Comb., 6, 1999. URL: http://www.combinatorics.org/Volume_6/Abstracts/v6i1r18.html.
  16. Sergei Evdokimov, Ilia Ponomarenko, and Gottfried Tinhofer. Forestal algebras and algebraic forests (on a new class of weakly compact graphs). Discrete Mathematics, 225(1-3):149-172, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00152-7.
  17. Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of graphs with small color classes by the Weisfeiler-Leman algorithm. Technical report, https://arxiv.org/abs/1907.02892, 2019. URL: https://arxiv.org/abs/1907.02892.
  18. Martin Fürer. On the combinatorial power of the Weisfeiler-Lehman algorithm. In Algorithms and Complexity - 10th International Conference (CIAC'17) Proceedings, volume 10236 of Lecture Notes in Computer Science, pages 260-271, 2017. URL: https://doi.org/10.1007/978-3-319-57586-5_22.
  19. François Le Gall. Powers of tensors and fast matrix multiplication. In Proc. of the Int. Symposium on Symbolic and Algebraic Computation (ISSAC'14), pages 296-303. ACM, 2014. URL: https://doi.org/10.1145/2608628.2608664.
  20. Branko Grünbaum. Configurations of points and lines, volume 103 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2009. Google Scholar
  21. Yuri Gurevich and Saharon Shelah. On finite rigid structures. J. Symb. Log., 61(2):549-562, 1996. URL: https://doi.org/10.2307/2275675.
  22. D.G. Higman. Finite permutation groups of rank 3. Math. Z., 86:145-156, 1964. URL: https://doi.org/10.1007/BF01111335.
  23. Oscar H. Ibarra, Shlomo Moran, and Roger Hui. A generalization of the fast LUP matrix decomposition algorithm and applications. J. Algorithms, 3(1):45-56, 1982. URL: https://doi.org/10.1016/0196-6774(82)90007-4.
  24. N. Immerman and E. Lander. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, pages 59-81. Springer, 1990. Google Scholar
  25. Neil Immerman. Descriptive complexity. Graduate texts in computer science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0539-5.
  26. Sandra Kiefer, Pascal Schweitzer, and Erkal Selman. Graphs identified by logics with counting. In Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS'15), volume 9234 of Lecture Notes in Computer Science, pages 319-330. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48057-1_25.
  27. Eugene M. Luks. Parallel algorithms for permutation groups and graph isomorphism. In 27th Annual Symposium on Foundations of Computer Science (FOCS'86), pages 292-302. IEEE Computer Society, 1986. URL: https://doi.org/10.1109/SFCS.1986.39.
  28. Klaus Metsch. Linear spaces with few lines, volume 1490 of Lect. Notes Math. Berlin etc.: Springer-Verlag, 1991. Google Scholar
  29. Mikhail Muzychuk and Ilya Ponomarenko. On quasi-thin association schemes. J. Algebra, 351(1):467-489, 2012. Google Scholar
  30. Daniel Neuen and Pascal Schweitzer. Benchmark graphs for practical graph isomorphism. In 25th Annual European Symposium on Algorithms (ESA'17), volume 87 of LIPIcs, pages 60:1-60:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.60.
  31. Daniel Neuen and Pascal Schweitzer. An exponential lower bound for individualization-refinement algorithms for graph isomorphism. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC'18), pages 138-150. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188900.
  32. Tomaž Pisanski and Brigitte Servatius. Configurations from a graphical viewpoint. Birkhäuser Adv. Texts, Basler Lehrbüch. New York, NY: Birkhäuser, 2013. Google Scholar
  33. Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4):17:1-17:24, 2008. URL: https://doi.org/10.1145/1391289.1391291.
  34. B.Yu. Weisfeiler and A.A. Leman. The reduction of a graph to canonical form and the algebra which appears therein. NTI, Ser. 2, 9:12-16, 1968. English translation is available at URL: https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf.
  35. Carme Álvarez and Birgit Jenner. A very hard log-space counting class. Theoretical Computer Science, 107(1):3-30, 1993. URL: https://doi.org/10.1016/0304-3975(93)90252-O.
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