The density maximization problem in graphs

  • Authors:

  • Mong-Jen Kao;Bastian Katz;Marcus Krug;D. T. Lee;Ignaz Rutter;Dorothea Wagner

  • Affiliations:

  • Dep. of Computer Science and Information Engineering, National Taiwan University, Taiwan;Faculty of Informatics, Karlsruhe Institute of Technology, Germany;Faculty of Informatics, Karlsruhe Institute of Technology, Germany;Dep. of Computer Science and Information Engineering, National Taiwan University, Taiwan;Faculty of Informatics, Karlsruhe Institute of Technology, Germany;Faculty of Informatics, Karlsruhe Institute of Technology, Germany

  • Venue:
  • COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
  • Year:
  • 2011

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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 we ∈ Z and edge lengths le ∈ N for e ∈ E we define the density of a pattern subgraph H = (V′, E′) ⊆ G as the ratio ∂(H) =Σe∈E′ we/Σe∈E′ le 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.