Abstract
Inferring graphs from path frequency has been studied as an important problem which has a potential application to drug design and elucidation of chemical structures. Given a multiple set g of strings of labels with length at most K, the problem asks to find a vertex-labeled graph G that attains a one-to-one correspondence between g and the occurrences of labels along all paths of length at most K in G. In this paper, we prove that the problem with K=1 can be formulated as a problem of finding a loopless and connected detachment, based on which an efficient algorithm for solving the problem is derived. Our algorithm also solves the problem with an additional constraint such that every vertex in an inferred graph is required to have a specified degree.
Similar content being viewed by others
References
Akutsu, T., Fukagawa, D.: Inferring a graph from path frequency. In: Proceedings of the 16th Annual Symposium on Combinatorial Pattern Matching. Lecture Notes in Computer Science, vol. 3537, pp. 371–382 (2005)
Akutsu, T., Fukagawa, D.: On inference of a chemical structure from path frequency. In: Proceedings of the 2005 International Joint Conference of InCoB, AASBi and KSBI, pp. 96–100 (2005)
Bakir, G.H., Weston, J., Schölkopf, B.: Learning to find pre-images. Adv. Neural Inf. Proc. Syst. 16, 449–456 (2004)
Bakir, G.H., Zien, A., Tsuda, K.: Learning to find graph pre-images. In: Proceedings of the 26th DAGM Symposium, Lecture Notes in Computer Science, vol. 3175, pp. 253–261 (2004)
Berg, A.R., Jackson, B., Jordán, T.: Highly edge-connected detachments of graphs and digraphs. J. Graph Theory 43, 67–77 (2003)
Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. Wiley–Interscience, New York (1998)
Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge (2000)
Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. 15, 948–957 (1986)
Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R.K., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon & Breach, New York (1970)
Frank, A.: A weighted matroid intersection algorithm. J. Algorithms 2, 328–336 (1981)
Fukunaga, T., Nagamochi, H.: Some theorems on detachments preserving local-edge-connectivity. In: Proceedings of the fifth CRACOW Conference on Graph Theory. Electronic Notes in Discrete Mathematics, vol. 24, pp. 173–180 (2006)
Jackson, B., Jordán, T.: Non-separable detachments of graphs. J. Comb. Theory B 87, 17–37 (2003)
Jansson, J., Sadakane, K.: Private communication
Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: Proceedings of the 20th International Conference on Machine Learning, pp. 321–328 (2003)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Berlin (2000)
Lawler, E.L.: Matroid intersection algorithms. Math. Program. 9, 31–56 (1975)
Nash-Williams, S.J.A.: Connected detachments of graphs and generalized Euler trails. J. Lond. Math. Soc. 31, 17–29 (1985)
Pólya, G.: Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nagamochi, H. A Detachment Algorithm for Inferring a Graph from Path Frequency. Algorithmica 53, 207–224 (2009). https://doi.org/10.1007/s00453-008-9184-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-008-9184-0