Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

  • Sepp Hartung
  • Christian Komusiewicz
  • André Nichterlein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7535)


Given an undirected graph G = (V,E) and an integer ℓ ≥ 1, the NP-hard 2-Club problem asks for a vertex set S ⊆ V of size at least ℓ such that the subgraph induced by S has diameter at most two. In this work, we extend previous parameterized complexity studies for 2-Club. On the positive side, we give polynomial kernels for the parameters “feedback edge set size of G” and “size of a cluster editing set of G” and present a direct combinatorial algorithm for the parameter “treewidth of G”. On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit a polynomial kernel with respect to the “size of a vertex cover of G”. Next, we show that, under the strong exponential time hypothesis, a previous O *(2|V| − ℓ) search tree algorithm [Schäfer et al., Optim. Lett. 2012] cannot be improved and that, unless NP ⊆ coNP/poly, there is no polynomial kernel for the dual parameter |V| − ℓ. Finally, we show that, in spite of this lower bound, the search tree algorithm for the dual parameter |V| − ℓ can be tuned into an efficient exact algorithm for 2-Club that substantially outperforms previous implementations.


Search Tree Vertex Cover Polynomial Kernel Marked Vertex Dual Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alba, R.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3(1), 113–126 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asahiro, Y., Miyano, E., Samizo, K.: Approximating Maximum Diameter-Bounded Subgraphs. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 615–626. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Balasundaram, B., Butenko, S., Trukhanovzu, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bourjolly, J.-M., Laporte, G., Pesant, G.: An exact algorithm for the maximum k-club problem in an undirected graph. European J. Oper. Res. 138(1), 21–28 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chang, M.S., Hung, L.J., Lin, C.R., Su, P.C.: Finding large k-clubs in undirected graphs. In: Proc. 28th Workshop on Combinatorial Mathematics and Computation Theory (2011)Google Scholar
  7. 7.
    Chen, Y., Flum, J., Müller, M.: Lower bounds for kernelizations and other preprocessing procedures. Theory Comput. Syst. 48(4), 803–839 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    DIMACS. Graph partitioning and graph clustering (2012), (accessed April 2012)
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  11. 11.
    Gendreau, M., Soriano, P., Salvail, L.: Solving the maximum clique problem using a tabu search approach. Ann. Oper. Res. 41(4), 385–403 (1993)zbMATHCrossRefGoogle Scholar
  12. 12.
    Hartung, S., Komusiewicz, C., Nichterlein, A.: On structural parameterizations for the 2-club problem (manuscript, June 2012) Google Scholar
  13. 13.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. System Sci. 63(4), 512–530 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS 105, 41–72 (2011)MathSciNetGoogle Scholar
  15. 15.
    Mahdavi, F., Balasundaram, B.: On inclusionwise maximal and maximum cardinality k-clubs in graphs. Discrete Optim. (to appear, 2012)Google Scholar
  16. 16.
    Memon, N., Larsen, H.L.: Structural Analysis and Mathematical Methods for Destabilizing Terrorist Networks Using Investigative Data Mining. In: Li, X., Zaïane, O.R., Li, Z.-h. (eds.) ADMA 2006. LNCS (LNAI), vol. 4093, pp. 1037–1048. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Mokken, R.J.: Cliques, Clubs and Clans. Quality and Quantity 13, 161–173 (1979)CrossRefGoogle Scholar
  18. 18.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  19. 19.
    Pasupuleti, S.: Detection of Protein Complexes in Protein Interaction Networks Using n-Clubs. In: Marchiori, E., Moore, J.H. (eds.) EvoBIO 2008. LNCS, vol. 4973, pp. 153–164. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5) (2012)Google Scholar
  21. 21.
    Schäfer, A.: Exact algorithms for s-club finding and related problems. Diploma thesis, Friedrich-Schiller-Universität Jena (2009)Google Scholar
  22. 22.
    Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sepp Hartung
    • 1
  • Christian Komusiewicz
    • 1
  • André Nichterlein
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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