Fast Algorithms for Diameter-Optimally Augmenting Paths

  • Ulrike Große
  • Joachim Gudmundsson
  • Christian Knauer
  • Michiel Smid
  • Fabian Stehn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


We consider the problem of augmenting a graph with \(n\) vertices embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present an exact algorithm for the cases when the input graph is a path that runs in \(O(n \log ^3 n)\) time. We also present an algorithm that computes a \((1+\varepsilon )\)-approximation in \(O(n + 1/\varepsilon ^3)\) time for paths in \({\mathbb {R}}^{d}\), where \(d\) is a constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Gyárfás, A., Ruszinkó, M.: Decreasing the diameter of bounded degree graphs. Journal of Graph Theory 35, 161–172 (1999)CrossRefGoogle Scholar
  2. 2.
    Bilò, D., Gualà, L., Proietti, G.: Improved approximability and non-approximability results for graph diameter decreasing problems. Theoretical Computer Science 417, 12–22 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to \(k\)-nearest-neighbors and \(n\)-body potential fields. Journal of the ACM 42, 67–90 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chepoi, V., Vaxès, Y.: Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica 33(2), 243–262 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chung, F.R.K., Garey, M.R.: Diameter bounds for altered graphs. Journal of Graph Theory 8(4), 511–534 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Dodis, Y., Khanna, S.: Designing networks with bounded pairwise distance. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC), pp. 750–759 (1999)Google Scholar
  7. 7.
    Erdős, P., Gyárfás, A., Ruszinkó, M.: How to decrease the diameter of triangle-free graphs. Combinatorica 18(4), 493–501 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Farshi, M., Giannopoulos, P., Gudmundsson, J.: Improving the stretch factor of a geometric network by edge augmentation. SIAM Journal on Computing 38(1), 226–240 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Frati, F., Gaspers, S., Gudmundsson, J., Mathieson, L.: Augmenting graphs to minimize the diameter. Algorithmica, 1–16 (2014)Google Scholar
  10. 10.
    Gao, Y., Hare, D.R., Nastos, J.: The parametric complexity of graph diameter augmentation. Discrete Applied Mathematics 161(10–11), 1626–1631 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ishii, T.: Augmenting outerplanar graphs to meet diameter requirements. Journal of Graph Theory 74, 392–416 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kapoor, S., Sarwat, M.: Bounded-diameter minimum-cost graph problems. Theory of Computing Systems 41(4), 779–794 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, C.-L., McCormick, S.T., Simchi-Levi, D.: On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems. Operations Research Letters 11(5), 303–308 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Luo, J., Wulff-Nilsen, C.: Computing best and worst shortcuts of graphs embedded in metric spaces. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 764–775. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  15. 15.
    Rutter, I., Wolff, A.: Augmenting the connectivity of planar and geometric graphs. Journal of Graph Algorithms and Applications 16(2), 599–628 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Diameter increase caused by edge deletion. Journal of Graph Theory 11, 409–427 (1997)CrossRefGoogle Scholar
  17. 17.
    Wulff-Nilsen, C.: Computing the dilation of edge-augmented graphs in metric spaces. Computational Geometry - Theory and Applications 43(2), 68–72 (2010)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ulrike Große
    • 1
  • Joachim Gudmundsson
    • 2
  • Christian Knauer
    • 1
  • Michiel Smid
    • 3
  • Fabian Stehn
    • 1
  1. 1.Institut für Angewandte InformatikUniversität BayreuthBayreuthGermany
  2. 2.School of Information TechnologyUniversity of SydneySydneyAustralia
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations