Abstract
In pattern recognition and data mining applications, where the underlying data is characterized by complex structural relationships, graphs are often used as a formalism for object representation. Yet, the high representational power and flexibility of graphs is accompanied by a significant increase of the complexity of many algorithms. For instance, exact computation of pairwise graph dissimilarity, i.e. distance, can be accomplished in exponential time complexity only. A previously introduced approximation framework reduces the problem of graph comparison to an instance of a linear sum assignment problem which allows graph dissimilarity computation in cubic time. The present paper introduces an extension of this approximation framework that runs in quadratic time. We empirically confirm the scalability of our extension with respect to the run time, and moreover show that the quadratic approximation leads to graph dissimilarities which are sufficiently accurate for graph based pattern classification.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Note that QAPs are known to be \(\mathcal {NP}\) -complete, and therefore, an exact and efficient algorithm for the graph edit distance problem can not be developed unless \(\mathcal {P} = \mathcal {NP}\).
- 2.
In [24] it is formally proven that this approximation scheme builds an upper bound of the exact graph edit distance.
- 3.
The assignment problem can also be formulated as finding a matching in a complete bipartite graph and is therefore also referred to as bipartite graph matching problem.
- 4.
References
Perner, P. (ed.): MLDM 2012. LNCS, vol. 7376. Springer, Heidelberg (2012)
Perner, P. (ed.): MLDM 2013. LNCS, vol. 7988. Springer, Heidelberg (2013)
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley-Interscience, New York (2000)
Bishop, C.: Pattern Recognition and Machine Learning. Springer, New York (2008)
Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004)
Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty years of graph matching in pattern recognition. Int. J. Pattern Recogn. Artif. Intell. 18(3), 265–298 (2004)
Foggia, P., Percannella, G., Vento, M.: Graph matching and learning in pattern recognition in the last 10 years. Int. J. Pattern Recogn. Artif. Intell. 28(1) (2014). http://dx.doi.org/10.1142/S0218001414500013
Cook, D., Holder, L. (eds.): Mining Graph Data. Wiley-Interscience, New York (2007)
Schenker, A., Bunke, H., Last, M., Kandel, A.: Graph-Theoretic Techniques for Web Content Mining. World Scientific Publishing, Singapore (2005)
Gärtner, T.: Kernels for Structured Data. World Scientific Publishng, Singapore (2008)
Gärtner, T., Horvath, T., Wrobel, S.: Graph kernels. In: Smmut, C., Webb, G.I. (eds.) Encyclopedia of Machine Learning, pp. 467–469. Springer US, London (2010)
Bunke, H., Allermann, G.: Inexact graph matching for structural pattern recognition. Pattern Recogn. Lett. 1, 245–253 (1983)
Sanfeliu, A., Fu, K.: A distance measure between attributed relational graphs for pattern recognition. IEEE Trans. Syst. Man Cybern. (Part B) 13(3), 353–363 (1983)
Cortés, X., Serratosa, F., Solé, A.: Active graph matching based on pairwise probabilities between nodes. In: Gimelfarb, G., Hancock, E., Imiya, A., Kuijper, A., Kudo, M. (eds.) SSPR 2012. LNCS, vol. 7626, pp. 98–106. Springer, Heidelberg (2012)
Boeres, M., Ribeiro, C., Bloch, I.: A randomized heuristic for scene recognition by graph matching. In: Ribeiro, C., Martins, S. (eds.) WEA 2004. LNCS, vol. 3059, pp. 100–113. Springer, Heidelberg (2004)
Sorlin, S., Solnon, C.: Reactive tabu search for measuring graph similarity. In: Brun, L., Vento, M. (eds.) GbRPR 2005. LNCS, vol. 3434, pp. 172–182. Springer, Heidelberg (2005)
Neuhaus, M., Bunke, H.: An error-tolerant approximate matching algorithm for attributed planar graphs and its application to fingerprint classification. In: Fred, A., Caelli, T.M., Duin, R.P.W., Campilho, A.C. (eds.) SSPR 2004. LNCS, vol. 3138, pp. 180–189. Springer, Heidelberg (2004)
Justice, D., Hero, A.: A binary linear programming formulation of the graph edit distance. IEEE Trans. Pattern Anal. Mach. Intell. 28(8), 1200–1214 (2006)
Dickinson, P., Bunke, H., Dadej, A., Kraetzl, M.: On graphs with unique node labels. In: Hancock, E., Vento, M. (eds.) GbRPR 2003. LNCS, vol. 2726, pp. 13–23. Springer, Heidelberg (2003)
Riesen, K., Bunke, H.: Approximate graph edit distance computation by means of bipartite graph matching. Image Vis. Comput. 27(4), 950–959 (2009)
Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Caetano, T.S., McAuley, J.J., Cheng, L., Le, Q.V., Smola, A.J.: Learning graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 31(6), 1048–1058 (2009)
Hart, P., Nilsson, N., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 4(2), 100–107 (1968)
Riesen, K., Fischer, A., Bunke, H.: Computing upper and lower bounds of graph edit distance in cubic time. Accepted for publication in Proceedings of the IAPR TC3 International Workshop on Artificial Neural Networks in Pattern Recognition
Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)
Kuhn, H.: The hungarian method for the assignment problem. Naval Res. Logistic Q. 2, 83–97 (1955)
Jonker, R., Volgenant, A.: A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38, 325–340 (1987)
Jonker, R., Volgenant, A.: Improving the hungarian assignment algorithm. Oper. Res. Lett. 5, 171–175 (1986)
Bertsekas, D.: The auction algorithm: a distributed relaxation method for the assignment problem. Ann. Oper. Res. 14, 105–123 (1988)
Hung, M.: A polynomial simplex method for the assignment problem. Oper. Res. 28, 969–982 (1983)
Orlin, J.: On the simplex algorithm for networks and generalized networks. Math. Program. Stud. 24, 166–178 (1985)
Ahuja, R., Orlin, J.: The scaling network simplex algorithm. Oper. Res. 40(1), 5–13 (1992)
Akgül, M.: A sequential dual simplex algorithm for the linear assignment problem. Oper. Res. Lett. 7, 155–518 (1988)
Srinivasan, V., Thompson, G.: Cost operator algorithms for the transportation problem. Math. Program. 12, 372–391 (1977)
Achatz, H., Kleinschmidt, P., Paparrizos, K.: A dual forest algorithm for the assignment problem. Appl. Geom. Discret. Math. AMS 4, 1–11 (1991)
Burkard, R., Ceia, E.: Linear assignment problems and extensions. Technical report 127, Karl-Franzens-Universität Graz und Technische Universität Graz (1998)
Avis, D.: A survey of heuristics for the weighted matching problem. Networks 13, 475–493 (1983)
Kurtzberg, J.: On approximation methods for the assignment problem. J. ACM 9(4), 419–439 (1962)
Riesen, K., Bunke, H.: Graph classification based on vector space embedding. Int. J. Pattern Recogn. Artif. Intell. 23(6), 1053–1081 (2008)
Neuhaus, M., Bunke, H.: Bridging the Gap Between Graph Edit Distance and Kernel Machines. World Scientific Publishing, Singapore (2007)
Riesen, K., Bunke, H.: IAM graph database repository for graph based pattern recognition and machine learning. In: da Vitoria Lobo, N., et al. (eds.) Structural, Syntactic, and Statistical Pattern Recognition. LNCS, vol. 5342, pp. 287–297. Springer, Heidelberg (2008)
Dosch, P., Valveny, E.: Report on the second symbol recognition contest. In: Wenyin, L., Lladós, J. (eds.) GREC 2005. LNCS, vol. 3926, pp. 381–397. Springer, Heidelberg (2005)
Acknowledgements
This work has been supported by the Hasler Foundation Switzerland and the Swiss National Science Foundation project 200021_153249.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Riesen, K., Ferrer, M., Dornberger, R., Bunke, H. (2015). Greedy Graph Edit Distance. In: Perner, P. (eds) Machine Learning and Data Mining in Pattern Recognition. MLDM 2015. Lecture Notes in Computer Science(), vol 9166. Springer, Cham. https://doi.org/10.1007/978-3-319-21024-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-21024-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21023-0
Online ISBN: 978-3-319-21024-7
eBook Packages: Computer ScienceComputer Science (R0)