International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 200-212 | Cite as

On Maximum Common Subgraph Problems in Series-Parallel Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

The complexity of the maximum common connected subgraph problem in partial k-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial 2-trees. On the contrary, the problem is known to be NP-hard in vertex-labeled partial 11-trees of bounded degree. We consider series-parallel graphs, i.e., partial 2-trees. We show that the problem remains NP-hard in biconnected series-parallel graphs with all but one vertex of degree bounded by three. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series-parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

Keywords

Maximum Common Subgraph Block and Bridge Preserving Series-parallel graphs 

References

  1. 1.
    Akutsu, T.: A polynomial time algorithm for finding a largest common subgraph of almost trees of bounded degree. IEICE Trans. Fundam. E76–A(9), 1488–1493 (1993)Google Scholar
  2. 2.
    Akutsu, T., Tamura, T.: On the complexity of the maximum common subgraph problem for partial k-trees of bounded degree. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 146–155. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  3. 3.
    Akutsu, T., Tamura, T.: A polynomial-time algorithm for computing the maximum common connected edge subgraph of outerplanar graphs of bounded degree. Algorithms 6(1), 119–135 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999) MATHCrossRefGoogle Scholar
  5. 5.
    Chimani, M., Hliněný, P.: A tighter insertion-based approximation of the crossing number. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 122–134. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. WH Freeman and Company, New York (1979) MATHGoogle Scholar
  7. 7.
    Gupta, A., Nishimura, N.: Sequential and parallel algorithms for embedding problems on classes of partial \(k\)-trees. In: Schmidt, E.M., Skyum, S. (eds.) SWAT 1994. LNCS, vol. 824. Springer, Heidelberg (1994) Google Scholar
  8. 8.
    Gupta, A., Nishimura, N.: The complexity of subgraph isomorphism for classes of partial \(k\)-trees. Theoret. Comput. Sci. 164(1–2), 287–298 (1996)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Horvth, T., Ramon, J.: Efficient frequent connected subgraph mining in graphs of bounded tree-width. Theoret. Comput. Sci. 411(3133), 2784–2797 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kriege, N., Mutzel, P.: Finding maximum common biconnected subgraphs in series-parallel graphs. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 505–516. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  11. 11.
    Kuhn, H.W.: The hungarian method for the assignment problem. Naval Res. Logistics Q. 2(1–2), 83–97 (1955)CrossRefGoogle Scholar
  12. 12.
    Kurpicz, F.: Efficient algorithms for the maximum common subgraph problem in partial 2-trees. Master’s thesis, TU Dortmund (2014)Google Scholar
  13. 13.
    Matouek, J., Thomas, R.: On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Math. 108(1–3), 343–364 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Matula, D.W.: Subtree isomorphism in \(O(n^{5/2})\). In: Algorithmic Aspects of Combinatorics, Ann. Discrete Math., vol. 2, pp. 91–106 (1978)Google Scholar
  15. 15.
    Nishizeki, T., Vygen, J., Zhou, X.: The edge-disjoint paths problem is np-complete for series-parallel graphs. Discrete Appl. Math. 115(1), 177–186 (2001)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Schietgat, L., Costa, F., Ramon, J., De Raedt, L.: Effective feature construction by maximum common subgraph sampling. Mach. Learn. 83(2), 137–161 (2011)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Schietgat, L., Ramon, J., Bruynooghe, M.: A polynomial-time metric for outerplanar graphs. In: Mining and Learning with Graphs (MLG) (2007)Google Scholar
  18. 18.
    Schietgat, L., Ramon, J., Bruynooghe, M., Blockeel, H.: An efficiently computable graph-based metric for the classification of small molecules. In: Boulicaut, J.-F., Berthold, M.R., Horváth, T. (eds.) DS 2008. LNCS (LNAI), vol. 5255, pp. 197–209. Springer, Heidelberg (2008) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnische Universität DortmundDortmundGermany

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