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)


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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  1. 1.
    Awerbuch, B.: Complexity of network synchronization. J. ACM 32(4), 804–823 (1985)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beauchamp, M.A.: An improved index of centrality. Behavioral Science 10, 161–163 (1965)CrossRefGoogle Scholar
  3. 3.
    Brandes, U., Erlebach, T. (eds.): Network Analysis. LNCS, vol. 3418. Springer, Heidelberg (2005)MATHGoogle Scholar
  4. 4.
    Brandes, U., Handke, D.: NP-completeness results for minimum planar spanners. Discr. Math. & Theor. Comp. Sci. 3(1), 1–10 (1998)MATHMathSciNetGoogle Scholar
  5. 5.
    Cai, L.: NP-completeness of minimum spanner problems. Discr. Appl. Math. 48(2), 187–194 (1994)MATHCrossRefGoogle Scholar
  6. 6.
    Camerini, P.M., Galbiati, G., Maffioli, F.: Complexity of spanning tree problems: Part I. Europ. J. Oper. Res. 5(5), 346–352 (1980)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comp. Sys. Sci. 39(2), 205–219 (1989)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dahlhaus, E., Dankelmann, P., Goddard, W., Swart, H.C.: MAD trees and distance-hereditary graphs. Discr. Appl. Math. 131(1), 151–167 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eckhardt, S., Kosub, S., Maaß, M.G., Täubig, H., Wernicke, S.: Combinatorial network abstraction by trees and distances. Technical Report TUM-I0502, Technische Universität München, Institut für Informatik (2005)Google Scholar
  10. 10.
    Elkin, M., Peleg, D.: (1 + ε,β)-spanner constructions for general graphs. SIAM J. Comp. 33(3), 608–631 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fekete, S.P., Kremer, J.: Tree spanners in planar graphs. Discr. Appl. Math. 108(1–2), 85–103 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hassin, R., Tamir, A.: On the minimum diameter spanning tree problem. Inform. Proc. Lett. 53(2), 109–111 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Johnson, D.S., Lenstra, J.K., Rinnooy Kan, A.H.G.: The complexity of the network design problem. Networks 8, 279–285 (1978)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kim, D.-H., Noh, J.D., Jeong, H.: Scale-free trees: The skeletons of complex networks. Phys. Rev. E 70(046126) (2004)Google Scholar
  16. 16.
    Kortsarz, G.: On the hardness of approximating spanners. Algorithmica 30(3), 432–450 (2001)MATHMathSciNetGoogle Scholar
  17. 17.
    Kratsch, D., Le, H.-O., Müller, H., Prisner, E., Wagner, D.: Additive tree spanners. SIAM J. Discr. Math. 17(2), 332–340 (2003)MATHCrossRefGoogle Scholar
  18. 18.
    Liestman, A.L., Shermer, T.C.: Additive graph spanners. Networks 23(4), 343–363 (1993)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Peleg, D., Reshef, E.: Low complexity variants of the arrow distributed directory. J. Comp. Sys. Sci. 63(3), 474–485 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comp. 18(4), 740–747 (1989)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Prisner, E.: Distance approximating spanning trees. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 499–510. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Sabidussi, G.: The centrality index of a graph. Psychometrica 31, 581–603 (1966)MATHCrossRefMathSciNetGoogle Scholar

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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