Advertisement

International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 1-12 | Cite as

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

  • Faisal N. Abu-Khzam
  • Édouard Bonnet
  • Florian SikoraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

Maximum Common Induced Subgraph (henceforth MCIS) is among the most studied classical \({\mathsf {NP}}\)-hard problems. MCIS remains \({\mathsf {NP}}\)-hard on many graph classes including bipartite graphs, planar graphs and k-trees. Little is known, however, about the parameterized complexity of the problem. When parameterized by the vertex cover number of the input graphs, the problem was recently shown to be fixed-parameter tractable. Capitalizing on this result, we show that the problem does not have a polynomial kernel when parameterized by vertex cover unless \({\mathsf {NP}}\subseteq \mathsf {coNP}/poly\). We also show that Maximum Common Connected Induced Subgraph (MCCIS), which is a variant where the solution must be connected, is also fixed-parameter tractable when parameterized by the vertex cover number of input graphs. Both problems are shown to be \({\mathsf {W[1]}}\)-complete on bipartite graphs and graphs of girth five and, unless \({\mathsf {P}}= {\mathsf {NP}}\), they do not belong to the class \({\mathsf {XP}}\) when parameterized by a bound on the size of the minimum feedback vertex sets of the input graphs, that is solving them in polynomial time is very unlikely when this parameter is a constant.

Notes

Acknowledgements

Work partially supported by the bilateral research cooperation CEDRE between France and Lebanon (grant number 30885TM).

References

  1. 1.
    Abu-Khzam, F.N.: Maximum common induced subgraph parameterized by vertex cover. Inf. Process. Lett. 114(3), 99–103 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Akutsu, T.: An RNC algorithm for finding a largest common subtree of two trees. IEICE Trans. Inf. Syst. 75(1), 95–101 (1992)Google Scholar
  3. 3.
    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
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28(1), 277–305 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cesati, M.: The turing way to parameterized complexity. J. Comput. Syst. Sci. 67(4), 654–685 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fellows, M.R., Jansen, B.M.P., Rosamond, F.A.: Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur. J. Comb. 34(3), 541–566 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput. 31(1), 113–145 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  11. 11.
    Grindley, H.M., Artymiuk, P.J., Rice, D.W., Willett, P.: Identification of tertiary structure resemblance in proteins using a maximal common subgraph isomorphism algorithm. J. Mol. Biol. 229(3), 707–721 (1993)CrossRefGoogle Scholar
  12. 12.
    Koch, I., Lengauer, T., Wanke, E.: An algorithm for finding maximal common subtopologies in a set of protein structures. J. Comput. Biol. 3(2), 289–306 (1996)CrossRefGoogle Scholar
  13. 13.
    McGregor, J., Willett, P.: Use of a maximal common subgraph algorithm in the automatic identification of the ostensible bond changes occurring in chemical reactions. J. Chem. Inf. Comput. Sci 21, 137–140 (1981)CrossRefGoogle Scholar
  14. 14.
    Moser, H., Sikdar, S.: The parameterized complexity of the induced matching problem. Discrete Appl. Math. 157(4), 715–727 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Raymond, J.W., Willett, P.: Maximum common subgraph isomorphism algorithms for the matching of chemical structures. J. Comput. Aided Mol. Des. 16, 521–533 (2002)CrossRefGoogle Scholar
  17. 17.
    Yamaguchi, A., Aoki, K.F., Mamitsuka, H.: Finding the maximum common subgraph of a partial k-tree and a graph with a polynomially bounded number of spanning trees. Inf. Process. Lett. 92(2), 57–63 (2004)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Faisal N. Abu-Khzam
    • 1
  • Édouard Bonnet
    • 2
  • Florian Sikora
    • 2
    Email author
  1. 1.Lebanese American UniversityBeirutLebanon
  2. 2.PSL, Université Paris-Dauphine, LAMSADE, UMR CNRS 7243ParisFrance

Personalised recommendations