On the Complexity of Frequent Subtree Mining in Very Simple Structures

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9046)

Abstract

We study the complexity of frequent subtree mining in very simple graphs beyond forests. We show for d-tenuous outerplanar graphs that frequent subtrees can be listed with polynomial delay if the cycle degree, i.e., the maximum number of blocks that share a common vertex, is bounded by some constant. The crucial step in the proof of this positive result is a polynomial time algorithm deciding subgraph isomorphism from trees into d-tenuous outerplanar graphs of bounded cycle degree. We obtain this algorithm by generalizing the algorithm of Shamir and Tsur that decides subgraph isomorphism between trees. Our results may also be of some interest to algorithmic graph theory, as they indicate that even for very simple structures, the cycle degree is a crucial parameter for the tractability of subgraph isomorphism. We also discuss some interesting problems towards generalizing the positive result of this work.

References

  1. 1.
    Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkamo, A.I.: Fast discovery of association rules. In: Advances in Knowledge Discovery and Data Mining, pp. 307–328. AAAI/MIT Press (1996)Google Scholar
  2. 2.
    Akutsu, T.: A polynomial time algorithm for finding a largest common subgraph of almost trees of bounded degree. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 76(9), 1488–1493 (1993)Google Scholar
  3. 3.
    Chi, Y., Yang, Y., Muntz, R.R.: Indexing and mining free trees. In: ICDM, pp. 509–512. IEEE Computer Society (2003)Google Scholar
  4. 4.
    De Raedt, L.: Logical and Relational Learning. Cognitive Technologies. Springer, Heidelberg (2008)CrossRefMATHGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory, vol. 173. Springer, Heidelberg (2012)MATHGoogle Scholar
  6. 6.
    Garriga, G.C., Khardon, R., De Raedt, L.: Mining closed patterns in relational, graph and network data. Ann. Math. Artif. Intell. 69(4), 315–342 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gottlob, G.: Subsumption and implication. Inf. Process. Lett. 24(2), 109–111 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hajiaghayi, M., Nishimura, N.: Subgraph isomorphism, log-bounded fragmentation, and graphs of (locally) bounded treewidth. J. Comput. Syst. Sci. 73(5), 755–768 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Han, J., Pei, J., Yin, Y., Mao, R.: Mining frequent patterns without candidate generation: a frequent-pattern tree approach. Data Min. Knowl. Discov. 8(1), 53–87 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Harary, F.: Graph Theory. Addison-Wesley Series in Mathematics. Perseus Books, Boulder (1994)MATHGoogle Scholar
  11. 11.
    Hopcroft, J.E., Karp, R.M.: An \({\rm n}^{\wedge }5/2\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Horváth, T., Ramon, J.: Efficient frequent connected subgraph mining in graphs of bounded tree-width. Theor. Comput. Sci. 411(31–33), 2784–2797 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Horváth, T., Ramon, J., Wrobel, S.: Frequent subgraph mining in outerplanar graphs. Data Min. Knowl. Discov. 21(3), 472–508 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Horváth, T., Turán, G.: Learning logic programs with structured background knowledge. Artif. Intell. 128(1–2), 31–97 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Matoušek, J., Thomas, R.: On the complexity of finding iso-and other morphisms for partial \(k\)-trees. Discrete Math. 108(1), 343–364 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Read, R.C., Tarjan, R.: Bound on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237–252 (1975)MathSciNetMATHGoogle Scholar
  18. 18.
    Shamir, R., Tsur, D.: Faster subtree isomorphism. In: Proceedings of the Fifth Israeli Symposium on the Theory of Computing and Systems, pp. 126–131. IEEE (1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pascal Welke
    • 1
  • Tamás Horváth
    • 1
    • 2
  • Stefan Wrobel
    • 1
    • 2
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany
  2. 2.Fraunhofer IAISSankt AugustinGermany

Personalised recommendations