Combinatorial Network Abstraction by Trees and Distances

  • Stefan Eckhardt
  • Sven Kosub
  • Moritz G. Maaß
  • Hanjo Täubig
  • Sebastian Wernicke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)

Abstract

This work draws attention to combinatorial network abstraction problems which are specified by a class  \(\mathcal{P}\) of pattern graphs and a real-valued similarity measure  \(\varrho\) based on certain graph properties. For fixed  \(\mathcal{P}\) and  \(\varrho\), the optimization task on any graph G is to find a subgraph G′ which belongs to  \(\mathcal{P}\) and minimizes \(\varrho(G,G^{\prime})\). We consider this problem for the natural case of trees and distance-based similarity measures. In particular, we systematically study spanning trees of graphs that minimize distances, approximate distances, and approximate closeness-centrality with respect to some standard vector and matrix norms. The complexity analysis shows that all considered variants of the problem are NP-complete, except for the case of distance-minimization with respect to the L ∞  norm. We further show that unless P = NP, there exist no polynomial-time constant-factor approximation algorithms for the distance-approximation problems if a subset of edges can be forced into the spanning tree.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Stefan Eckhardt
    • 1
  • Sven Kosub
    • 1
  • Moritz G. Maaß
    • 1
  • Hanjo Täubig
    • 1
  • Sebastian Wernicke
    • 2
  1. 1.Fakultät für InformatikTechnische Universität MünchenGarchingGermany
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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