Document

Graph Similarity and Approximate Isomorphism

File

LIPIcs.MFCS.2018.20.pdf
• Filesize: 478 kB
• 16 pages

Cite As

Martin Grohe, Gaurav Rattan, and Gerhard J. Woeginger. Graph Similarity and Approximate Isomorphism. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.20

Abstract

The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery.

Subject Classification

ACM Subject Classification
• Mathematics of computing → Graph algorithms
Keywords
• Graph Similarity
• Approximate Graph Isomorphism

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. H.A. Almohamad and S.O. Duffuaa. A linear programming approach for the weighted graph matching problem. IEEE Transactions on pattern analysis and machine intelligence, 15(5):522-525, 1993.
2. S. Arora, A. Frieze, and H. Kaplan. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Mathematical programming, 92(1):1-36, 2002.
3. V. Arvind, J. Köbler, S. Kuhnert, and Y. Vasudev. Approximate graph isomorphism. In B. Rovan, V. Sassone, and P. Widmayer, editors, Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, volume 7464 of Lecture Notes in Computer Science, pages 100-111. Springer Verlag, 2012.
4. L. Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC '16), pages 684-697, 2016.
5. R.E. Burkard, E. Cela, G. Rote, and G.J. Woeginger. The quadratic assignment problem with a monotone anti-monge and a symmetric toeplitz matrix: easy and hard cases. Mathematical Programming, 82:125-158, 1998.
6. E. Cela. The Quadratic Assignment Problem: Theory and Algorithms. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
7. E. Cela, V.G. Deineko, and G.J. Woeginger. Well-solvable cases of the qap with block-structured matrices. Discrete Applied Mathematics, 186:56-65, 2015.
8. E. Cela, N. Schmuck, S. Wimer, and G.J. Woeginger. The wiener maximum quadratic assignment problem. Discrete Optimization, 8:411-416, 2011.
9. P. Codenotti, H. Katebi, K. A. Sakallah, and I. L. Markov. Conflict analysis and branching heuristics in the search for graph automorphisms. In 2013 IEEE 25th International Conference on Tools with Artificial Intelligence, Herndon, VA, USA, November 4-6, 2013, pages 907-914, 2013.
10. D. Conte, P. Foggia, C. Sansone, and M. Vento. Thirty years of graph matching in pattern recognition. International journal of pattern recognition and artificial intelligence, 18(3):265-298, 2004.
11. A.N. Elshafei. Hospital layout as a quadratic assignment problem. Operational Research Quarterly, 28:167-179, 1977.
12. M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.
13. A.M. Geoffrion and G.W. Graves. Scheduling parallel production lines with changeover costs: Practical application of a quadratic assignment/lp approach. Operational Research, 24:595-610, 1976.
14. S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Transactions on pattern analysis and machine intelligence, 18(4):377-388, 1996.
15. T. Junttila and P. Kaski. Engineering an efficient canonical labeling tool for large and sparse graphs. In Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithms and Combinatorics, pages 135-149. SIAM, 2007.
16. P. Keldenich. Random robust graph isomorphism. Master’s thesis, Department of Compter Science, RWTH Aachen University, 2015.
17. Y. Koren. Drawing graphs by eigenvectors: theory and practice. Computers and Mathematics with Applications, 49(11):1867-1888, 2005.
18. J. Krarup and Pruzan P.M. Computer-aided layout design. Mathematical Programming Study, 9:75-94, 1978.
19. K. Makarychev, R. Manokaran, and M. Sviridenko. Maximum quadratic assignment problem: Reduction from maximum label cover and lp-based approximation algorithm. ACM Transactions on Algorithms, 10(4):18, 2014.
20. D.W. Matula. Subtree isomorphism in o(n^5/2). In P. H. B. Alspach and D. Miller, editors, Algorithmic Aspects of Combinatorics, volume 2 of Annals of Discrete Mathematics, pages 91-106. Elsevier, 1978.
21. B. McKay. Practical graph isomorphism. Congressus Numerantium, 30:45-87, 1981.
22. B. D. McKay and A. Piperno. Practical graph isomorphism, II. J. Symb. Comput., 60:94-112, 2014.
23. S. Melnik, H. Garcia-Molina, and E. Rahm. Similarity flooding: A versatile graph matching algorithm and its application to schema matching. In Proceedings. 18th International Conference on Data Engineering, pages 117-128, 2002.
24. V. Nagarajan and M. Sviridenko. On the maximum quadratic assignment problem. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 516-524, 2009.
25. D. Neuen and P. Schweitzer. Benchmark graphs for practical graph isomorphism. ArXiv (CoRR), arXiv:1705.03686 [cs.DS], 2017.
26. R. O'Donnell, J. Wright, C. Wu, and Y. Zhou. Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1659-1677, 2014.
27. M.A. Pollatschek, N. Gershoni, and Y.T. Radday. Optimization of the typewriter keyboard by simulation. Angewandte Informatik, 17:438-439, 1976.
28. F. Rendl and H. Wolkowicz. Applications of parametric programming and Eigenvalue maximization to the quadratic assignment problem. Mathematical Programming, 53:63-78, 1992.
29. S. Umeyama. An eigendecomposition approach to weighted graph matching problems. IEEE transactions on pattern analysis and machine intelligence, 10(5):695-703, 1988.
30. M. Zaslavskiy, F. Bach, and J.-P. Vert. A path following algorithm for the graph matching problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(12):2227-2242, 2009.
X

Feedback for Dagstuhl Publishing