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The Density Maximization Problem in Graphs

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Computing and Combinatorics (COCOON 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6842))

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Abstract

We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V,E) with edge weights w e  ∈ ℤ and edge lengths ℓ e  ∈ ℕ for e ∈ E we define the density of a pattern subgraph H = (V′,E′) ⊆ G as the ratio \(\ensuremath{\varrho}(H)=\sum_{e\in E'}w_e/\sum_{e\in E'} \ell_e\). We consider the problem of computing a maximum density pattern H with weight at least W and and length at most L in a host G.

We consider this problem for different classes of hosts and patterns. We show that it is NP-hard even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty.

Supported by NSC-DFG Projects NSC98-2221-E-001-007-MY3 and WA 654/18.

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Author information

Authors and Affiliations

  1. Faculty of Informatics, Karlsruhe Institute of Technology (KIT), Germany

    Bastian Katz, Marcus Krug, Ignaz Rutter & Dorothea Wagner

  2. Dep. of Computer Science and Information Engineering, National Taiwan University, Taiwan

    Mong-Jen Kao & D. T. Lee

Authors
  1. Mong-Jen Kao
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  2. Bastian Katz
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  3. Marcus Krug
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  4. D. T. Lee
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  5. Ignaz Rutter
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  6. Dorothea Wagner
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas-Pan American, 78539, Edinburg, TX, USA

    Bin Fu

  2. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Ding-Zhu Du

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© 2011 Springer-Verlag Berlin Heidelberg

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Kao, MJ., Katz, B., Krug, M., Lee, D.T., Rutter, I., Wagner, D. (2011). The Density Maximization Problem in Graphs. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-22685-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22684-7

  • Online ISBN: 978-3-642-22685-4

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