Finding Dense Subgraphs of Sparse Graphs

  • Christian Komusiewicz
  • Manuel Sorge
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7535)

Abstract

We investigate the computational complexity of the Densest-k-Subgraph (DkS) problem, where the input is an undirected graph G = (V,E) and one wants to find a subgraph on exactly k vertices with a maximum number of edges. We extend previous work on DkS by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, DkS becomes fixed-parameter tractable with respect to either of the parameters maximum degree and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for DkS with respect to the combined parameter “degeneracy of G and |V| − k”. On the negative side, we find that DkS is W[1]-hard with respect to the combined parameter “solution size k and degeneracy of G”. We furthermore strengthen a previous hardness result for DkS [Cai, Comput. J., 2008] by showing that for every fixed μ, 0 < μ < 1, the problem of deciding whether G contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V| − k.

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References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Amini, O., Sau, I., Saurabh, S.: Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 13–29. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: The maximum k-plex problem. Oper. Res. 59(1), 133–142 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proc. 28th STACS. LIPIcs, vol. 9, pp. 165–176. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  5. 5.
    Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Comput. J. 51(1), 102–121 (2008)CrossRefGoogle Scholar
  6. 6.
    Cai, L., Chan, S.M., Chan, S.O.: Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  8. 8.
    Eppstein, D., Spiro, E.S.: The h-Index of a Graph and Its Application to Dynamic Subgraph Statistics. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 278–289. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Eppstein, D., Löffler, M., Strash, D.: Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 403–414. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Feige, U., Seltser, M.: On the densest k-subgraph problem. Technical report, The Weizmann Institute, Department of Applied Math. and Computer Science (1997)Google Scholar
  11. 11.
    Feige, U., Peleg, D., Kortsarz, G.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Holzapfel, K., Kosub, S., Maaß, M.G., Täubig, H.: The complexity of detecting fixed-density clusters. Discrete Appl. Math. 154(11), 1547–1562 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Khuller, S., Saha, B.: On Finding Dense Subgraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 597–608. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Komusiewicz, C.: Parameterized Algorithmics for Network Analysis: Clustering & Querying. PhD thesis, Technische Universität Berlin, Berlin, Germany (2011)Google Scholar
  17. 17.
    Kosub, S.: Local Density. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 112–142. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS 105, 41–72 (2011)MathSciNetGoogle Scholar
  19. 19.
    Saha, B., Hoch, A., Khuller, S., Raschid, L., Zhang, X.-N.: Dense Subgraphs with Restrictions and Applications to Gene Annotation Graphs. In: Berger, B. (ed.) RECOMB 2010. LNCS, vol. 6044, pp. 456–472. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christian Komusiewicz
    • 1
  • Manuel Sorge
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany

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