article

Graph Isomorphism, Color Refinement, and Compactness

Authors:

Johannes Köbler,

Oleg VerbitskyAuthors Info & Claims

Computational Complexity, Volume 26, Issue 3

Pages 627 - 685

https://doi.org/10.1007/s00037-016-0147-6

Published: 01 September 2017 Publication History

Abstract

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph Gamenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai et al. (SIAM J Comput 9(3):628---635, 1980) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time $${O((n+m)\log n)}$$O((n+m)logn), where n and m denote the number of vertices and the number of edges in the input graph.

We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs. A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2---3):253---264, 1991) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: źWe show that all amenable graphs are compact. In other words, the applicability range for Tinhofer's linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement.źExploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy, and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.

References

[1]

Dana Angluin (1980). Local and global properties in networks of processors. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, 82-93. ACM.

[2]

V. Arvind, Johannes Köbler, Gaurav Rattan & Oleg Verbitsky (2015a). On the power of color refinement. In Proceedings of the 20th International Symposium on Fundamentals of Computation Theory (FCT), volume 9210 of Lecture Notes in Computer Science, 339-350. Springer.

[3]

V. Arvind, Johannes Köbler, Gaurav Rattan & Oleg Verbitsky (2015b). On Tinhofer's linear programming approach to isomorphism testing. In Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 9235 of Lecture Notes in Computer Science, 26-37. Springer.

[4]

Albert Atserias & Elitza N. Maneva (2013). Sherali-Adams relaxations and indistinguishability in counting logics. SIAM Journal on Computing 42(1), 112-137.

[5]

László Babai (1995). Automorphism groups, isomorphism, reconstruction. In Handbook of Combinatorics, chapter 27, 1447-1540. Elsevier.

Digital Library

[6]

László Babai, Paul Erd?os & Stanley M. Selkow (1980). Random graph isomorphism. SIAM Journal on Computing 9(3), 628-635.

Digital Library

[7]

László Babai & Ludek Ku?cera (1979). Canonical labelling of graphs in linear average time. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, 39-46.

[8]

Christoph Berkholz, Paul Bonsma & Martin Grohe (2013). Tight lower and upper bounds for the complexity of canonical colour refinement. In Proceedings of 21st Annual European Symposium on Algorithms (ESA), volume 8125 of Lecture Notes in Computer Science, 145-156. Springer.

[9]

Alessandro Borri, Tiziana Calamoneri & Rossella Petreschi (2011). Recognition of unigraphs through superposition of graphs. Journal of Graph Algorithms and Applications 15(3), 323-343.

[10]

Richard A. Brualdi (1988). Some applications of doubly stochastic matrices. Linear Algebra and its Applications 107, 77-100.

[11]

Richard A. Brualdi (2006). Combinatorial matrix classes. Cambridge University Press.

[12]

Robert G. Busacker & Thomas L. Saaty (1965). Finite graphs and networks: an introduction with applications. International Series in Pure and Applied Mathematics. McGraw-Hill Book Company, New York etc.

[13]

Jin-Yi Cai, Martin Fürer & Neil Immerman (1992). An optimal lower bound on the number of variables for graph identification. Combinatorica 12(4), 389-410.

[14]

A. Cardon & Maxime Crochemore (1982). Partitioning a graph in O(|A| log₂|V|). Theoretical Computer Science 19, 85-98.

[15]

Ada Chan & Chris D. Godsil (1997). Symmetry and eigenvectors. In Graph symmetry, volume 497, 75-106. Kluwer Acad. Publ., Dordrecht.

[16]

Sergei Evdokimov, Marek Karpinski & Ilia N. Ponomarenko (1999). Compact cellular algebras and permutation groups. Discrete Mathematics 197-198, 247-267.

Digital Library

[17]

Sergei Evdokimov, Ilia N. Ponomarenko & Gottfried Tinhofer (2000). Forestal algebras and algebraic forests (on a new class of weakly compact graphs). Discrete Mathematics 225(1-3), 149-172.

[18]

Chris D. Godsil (1997). Compact graphs and equitable partitions. Linear Algebra and its Applications 255(1-3), 259-266.

[19]

Leslie M. Goldschlager (1977). The monotone and planar circuit value problems are log space complete for P. SIGACT News 9, 25-29.

Digital Library

[20]

Martin Grohe (1999). Equivalence in finite-variable logics is complete for polynomial time. Combinatorica 19(4), 507-532.

[21]

Martin Grohe, Kristian Kersting, Martin Mladenov & Erkal Selman (2014). Dimension reduction via colour refinement. In Proceeding of 22th Annual European Symposium on Algorithms (ESA), volume 8737 of Lecture Notes in Computer Science, 505-516. Springer.

[22]

Martin Grohe & Martin Otto (2015). Pebble games and linear equations. The Journal of Symbolic Logic 80(3), 797-844.

[23]

Neil Immerman & Eric Lander (1990). Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, 59-81. Springer.

[24]

Robert H. Johnson (1975). Simple separable graphs. Pacific Journal of Mathematics 56, 143-158.

[25]

Kristian Kersting, Martin Mladenov, Roman Garnett & Martin Grohe (2014). Power iterated color refinement. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, 1904-1910. AAAI Press.

[26]

Sandra Kiefer, Pascal Schweitzer & Erkal Selman (2015). Graphs identified by logics with counting. In Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 9234 of Lecture Notes in Computer Science, 319-330. Springer. A full version is available as an e-print http://arxiv.org/abs/1503.08792.

[27]

Johannes Köbler, Uwe Schöning & Jacobo Torán (1993). The graph isomorphism problem: its structural complexity. Boston, MA: Birkhäuser.

[28]

Michael Koren (1976). Pairs of sequences with a unique realization by bipartite graphs. Journal of Combinatorial Theory, Series B 21(3), 224-234.

[29]

Andreas Krebs & Oleg Verbitsky (2015). Universal covers, color refinement, and two-variable logic with counting quantifiers: Lower bounds for the depth. In Proceedings of the 30-th ACM/IEEE Annual Symposium on Logic in Computer Science (LICS), 689-700. IEEE Computer Society.

[30]

Frank Thomson Leighton (1982). Finite common coverings of graphs. Journal of Combinatorial Theory, Series B 33(3), 231-238.

[31]

Peter N. Malkin (2014). Sherali-Adams relaxations of graph isomorphism polytopes. Discrete Optimization 12, 73-97.

[32]

Motakuri V. Ramana, Edward R. Scheinerman & Daniel Ullman (1994). Fractional isomorphism of graphs. Discrete Mathematics 132(1-3), 247-265.

Digital Library

[33]

Helmut Schreck & Gottfried Tinhofer (1988). A note on certain subpolytopes of the assignment polytope associated with circulant graphs. Linear Algebra and its Applications 111, 125-134.

[34]

Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn & Karsten M. Borgwardt (2011). Weisfeiler-Lehman graph kernels. Journal of Machine Learning Research 12, 2539-2561.

Digital Library

[35]

Gottfried Tinhofer (1986). Graph isomorphism and theorems of Birkhoff type. Computing 36, 285-300.

Digital Library

[36]

Gottfried Tinhofer (1989). Strong tree-cographs are Birkhoff graphs. Discrete Applied Mathematics 22(3), 275-288.

Digital Library

[37]

Gottfried Tinhofer (1991). A note on compact graphs. Discrete Applied Mathematics 30(2-3), 253-264.

Digital Library

[38]

Gottfried Tinhofer & Mikhail Klin (1999). Algebraic combinatorics in mathematical chemistry. Methods and algorithms III. Graph invariants and stabilization methods. Technical Report TUM-M9902, Technische Universität München.

[39]

Regina Tyshkevich (2000). Decomposition of graphical sequences and unigraphs. Discrete Mathematics 220(1-3), 201-238.

Digital Library

[40]

Gabriel Valiente (2002). Algorithms on trees and graphs. Springer.

[41]

Ping Wang & Jiong Sheng Li (2005). On compact graphs. Acta Mathematica Sinica 21(5), 1087-1092.

[42]

Boris Yu. Weisfeiler & Andrei A. Leman (1968). A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, Seriya 22 9, 12-16. In Russian.

[43]

Hassler Whitney (1932). Congruent graphs and connectivity of graphs. American Journal of Mathematics 54, 150-168.

[44]

Matan Ziv-Av (2013). Results of computer algebra calculations for triangle free strongly regular graphs. E-print http://www.math.bgu.ac.il/~zivav/math/eqpart.pdf.

Cited By

Gavrilyuk ANedela RPonomarenko I(2023)The Weisfeiler–Leman Dimension of Distance-Hereditary GraphsGraphs and Combinatorics10.1007/s00373-023-02683-339:4Online publication date: 14-Jul-2023
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Kriege NKoyejo SMohamed SAgarwal ABelgrave DCho KOh A(2022)Weisfeiler and leman go walkingProceedings of the 36th International Conference on Neural Information Processing Systems10.5555/3600270.3601733(20119-20132)Online publication date: 28-Nov-2022
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Ponomarenko IRyabov G(2022)On pseudofrobenius imprimitive association schemesJournal of Algebraic Combinatorics: An International Journal10.1007/s10801-022-01193-457:2(385-402)Online publication date: 29-Dec-2022
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Index Terms

Graph Isomorphism, Color Refinement, and Compactness
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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cover image Computational Complexity

Computational Complexity Volume 26, Issue 3

September 2017

230 pages

ISSN:1016-3328

Issue’s Table of Contents

Copyright © Copyright © 2017 Springer International Publishing AG.

Publisher

Birkhauser Verlag

Switzerland

Publication History

Published: 01 September 2017

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Cited By

Gavrilyuk ANedela RPonomarenko I(2023)The Weisfeiler–Leman Dimension of Distance-Hereditary GraphsGraphs and Combinatorics10.1007/s00373-023-02683-339:4Online publication date: 14-Jul-2023
https://dl.acm.org/doi/10.1007/s00373-023-02683-3
Kriege NKoyejo SMohamed SAgarwal ABelgrave DCho KOh A(2022)Weisfeiler and leman go walkingProceedings of the 36th International Conference on Neural Information Processing Systems10.5555/3600270.3601733(20119-20132)Online publication date: 28-Nov-2022
https://dl.acm.org/doi/10.5555/3600270.3601733
Ponomarenko IRyabov G(2022)On pseudofrobenius imprimitive association schemesJournal of Algebraic Combinatorics: An International Journal10.1007/s10801-022-01193-457:2(385-402)Online publication date: 29-Dec-2022
https://dl.acm.org/doi/10.1007/s10801-022-01193-4
Kiefer SSchweitzer PSelman E(2021)Graphs Identified by Logics with CountingACM Transactions on Computational Logic10.1145/341751523:1(1-31)Online publication date: 22-Oct-2021
https://dl.acm.org/doi/10.1145/3417515
Bildanov RPanshin VRyabov G(2021)On WL-Rank and WL-Dimension of Some Deza Circulant GraphsGraphs and Combinatorics10.1007/s00373-021-02364-z37:6(2397-2421)Online publication date: 1-Nov-2021
https://dl.acm.org/doi/10.1007/s00373-021-02364-z
Ponomarenko IRyabov G(2021)The Weisfeiler–Leman Dimension of Chordal Bipartite Graphs Without Bipartite ClawGraphs and Combinatorics10.1007/s00373-021-02308-737:3(1089-1102)Online publication date: 1-May-2021
https://dl.acm.org/doi/10.1007/s00373-021-02308-7
Ok SLarochelle HRanzato MHadsell RBalcan MLin H(2020)A graph similarity for deep learningProceedings of the 34th International Conference on Neural Information Processing Systems10.5555/3495724.3495725(1-12)Online publication date: 6-Dec-2020
https://dl.acm.org/doi/10.5555/3495724.3495725
Kiefer S(2020)The Weisfeiler-Leman algorithmACM SIGLOG News10.1145/3436980.34369827:3(5-27)Online publication date: 16-Nov-2020
https://dl.acm.org/doi/10.1145/3436980.3436982

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