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Finding Dense Subgraphs of Sparse Graphs

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Parameterized and Exact Computation (IPEC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7535))

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Abstract

We investigate the computational complexity of the Densest- k -Subgraph (D k S) 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 D k S by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, D k S 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 D k S with respect to the combined parameter “degeneracy of G and |V| − k”. On the negative side, we find that D k S 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 D k S [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.

One result of this work (Thm. 4) is contained in the first author’s dissertation [16].

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany

    Christian Komusiewicz & Manuel Sorge

Authors
  1. Christian Komusiewicz
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  2. Manuel Sorge
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Editors and Affiliations

  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784, Athens, Greece

    Dimitrios M. Thilikos

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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Komusiewicz, C., Sorge, M. (2012). Finding Dense Subgraphs of Sparse Graphs. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_23

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  • DOI: https://doi.org/10.1007/978-3-642-33293-7_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33292-0

  • Online ISBN: 978-3-642-33293-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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