Augmenting Local Edge-Connectivity between Vertices and Vertex Subsets in Undirected Graphs

  • Toshimasa Ishii
  • Masayuki Hagiwara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)


Given an undirected multigraph G=(V,E), a family \({\cal W}\) of sets W ⊆ V of vertices (areas), and a requirement function \({r_{{\cal W}}} : {\cal W} \)Z  +  (where Z  +  is the set of positive integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least \({r_{{\cal W}}}(W)\) edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area \(W \in {\cal W}\). So far this problem was shown to be NP-hard in the uniform case of \({r_{{\cal W}}}(W)=1\) for each \(W \in {\cal W}\), and polynomially solvable in the uniform case of \({r_{{\cal W}}}(W)=r \geq 2\) for each \(W \in {\cal W}\). In this paper, we show that the problem can be solved in O(m + pr * n 5 log(n/r *)) time, even in the general case of \({r_{{\cal W}}}(W)\geq 3\) for each \(W \in {\cal W}\), where n=|V|, m = |{{u,v}| (u,v) ∈ E}|, \(p=|{\cal W}|\), and \(r^*=\max\{{r_{{\cal W}}}(W)\mid W \in {\cal W}\}\). Moreover, we give an approximation algorithm which finds a solution with at most one surplus edges over the optimal value in the same time complexity in the general case of \({r_{{\cal W}}}(W)\geq 2\) for each \(W \in {\cal W}\).


Admissible Pair Requirement Function Uniform Case Area Graph Vertex Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math. 5(1), 25–53 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Frank, A.: Connectivity augmentation problems in network design. In: Birge, J.R., Murty, K.G. (eds.) Mathematical Programming: State of the Art 1994, pp. 34–63. The University of Michigan, Ann Arbor (1994)Google Scholar
  3. 3.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. Assoc. Comput. Mach. 35, 921–940 (1988)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Ishii, T., Akiyama, Y., Nagamochi, H.: Minimum augmentation of edge-connectivity between vertices and sets of vertices in undirected graphs. In: Computing Theory: The Australian Theory Symposium (CATS 2003). Electr. Notes Theo. Comp. Sci., vol. 78 (2003)Google Scholar
  5. 5.
    Ito, H.: Node-to-area connectivity of graphs. Transactions of the Institute of Electrical Engineers of Japan 11C(4), 463–469 (1994)Google Scholar
  6. 6.
    Ito, H.: Node-to-area connectivity of graphs. In: Fushimi, M., Tone, K. (eds.) Proceedings of APORS 1994, pp. 89–96. World Scientific publishing, Singapore (1995)Google Scholar
  7. 7.
    Ito, H., Yokoyama, M.: Edge connectivity between nodes and node-subsets. Networks 31(3), 157–164 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jordán, T.: Two NP-complete augmentation problems. Preprint no. 8, Department of Mathematics and Computer Science. Odense University (1997)Google Scholar
  9. 9.
    Miwa, H., Ito, H.: Edge augmenting problems for increasing connectivity between vertices and vertex subsets, 1999 Technical Report of IPSJ, 99-AL-66(8), 17–24 (1999) Google Scholar
  10. 10.
    Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Nagamochi, H., Ibaraki, T.: Computing edge-connectivity of multigraphs and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nagamochi, H., Ibaraki, T.: Graph connectivity and its augmentation: applications of MA orderings. Discrete Applied Mathematics 123(1), 447–472 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. J. Comput. System Sci. 35, 96–144 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Toshimasa Ishii
    • 1
  • Masayuki Hagiwara
    • 1
  1. 1.Department of Information and Computer SciencesToyohashi University of TechnologyAichiJapan

Personalised recommendations