Combinatorial network abstraction by trees and distances

  • Authors:

  • Stefan Eckhardt;Sven Kosub;Moritz G. Maaß;Hanjo Täubig;Sebastian Wernicke

  • Affiliations:

  • Fakultät für Informatik, Technische Universität München, Garching, Germany;Fakultät für Informatik, Technische Universität München, Garching, Germany;Fakultät für Informatik, Technische Universität München, Garching, Germany;Fakultät für Informatik, Technische Universität München, Garching, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany

  • Venue:
  • ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
  • Year:
  • 2005

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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.