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Approximation algorithms for graph augmentation

  • Samir Khuller
  • Ramakrishna Thurimella
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 623)

Abstract

We study the problem of increasing the connectivity1 of a graph at an optimal cost. Since the general problem is NP-hard, we focus on efficient approximation schemes that come within a constant factor from the optimal. Previous algorithms either do not take edge costs into consideration, or run slower than our algorithm. Our algorithm takes as input an undirected graph G0 = (V, E0) on n vertices, that is not necessarily connected, and a set Feasible of m weighted edges on V, and outputs a subset Aug of edges which when added to G0 make it 2-connected. The weight of Aug, when G0 is initially connected, is no more than twice the weight of the least weight subset of edges of Feasible that increases the connectivity to 2. The running time of our algorithm is O(m + n logn). We also study the problem of increasing the edge connectivity of any graph G, to k, within a factor of 2 (for any k > 0). The running time of this algorithm is O(nk log n(m + n log n)). We observe that when k is odd we can use different techniques to obtain an approximation factor of 2 for increasing edge connectivity from k to (k+1) in O(kn2) time.

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References

  1. [AHU]
    A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms, Addison-Wesley, 1974.Google Scholar
  2. [DKL76]
    E.A. Dinits, A.V. Karzanov and M.L. Lomosonov, “On the structure of a family of minimal weighted cuts in a graph,” Studies in Discrete Optimization [In Russian], A.A. Fridman (ear decomposition), Nauka, Moscow, pp. 290–306, 1976.Google Scholar
  3. [NGM90]
    D. Naor, D. Gusfield and C. Martel, “A fast algorithm for optimally increasing the edge-connectivity,” 31st Annual Symposium on Foundations of Computer Science, pp. 698–707, (1990).Google Scholar
  4. [Ed79]
    J. Edmonds, “Matroid intersection,” Annals of Discrete Mathematics, No. 4, pp. 185–204, (1979).Google Scholar
  5. [Ev79]
    S. Even, Graph Algorithms, Computer Science Press, Potomac, Md., 1979.Google Scholar
  6. [ET76]
    K. P. Eswaran and R. E. Tarjan, “Augmentation problems,” SIAM Journal on Computing, Vol. 5, No. 4, pp. 653–665, (1976).CrossRefGoogle Scholar
  7. [Fr90]
    A. Frank, “Augmenting graphs to meet edge-connectivity requirements,” 31st Annual Symposium on Foundations of Computer Science, pp. 708–718, (1990).Google Scholar
  8. [FJ81]
    G. N. Frederickson and J. JáJá, “Approximation algorithms for several graph augmentation problems,” SIAM Journal on Computing, Vol. 10, No. 2, pp. 270–283, (1981).CrossRefGoogle Scholar
  9. [FJ82]
    G. N. Frederickson and J. JáJá, On the relationship between the biconnectivity augmentation and traveling salesman problems,” Theoretical Computer Science, Vol. 19, No. 2, pp. 189–201, (1982).CrossRefGoogle Scholar
  10. [FT89]
    A. Frank and E. Tardos, “An application of submodular flows,” Linear Algebra and its Applications, 114/115, pp. 320–348, (1989).Google Scholar
  11. [G91a]
    H. N. Gabow, “A matroid approach to finding edge connectivity and packing arborescences,” 23rd Annual Symposium on Theory of Computing, pp. 112–122, (1991).Google Scholar
  12. [G91b]
    H. N. Gabow, “Applications of a poset representation to edge connectivity and graph rigidity,” 32nd Annual Symposium on Foundations of Computer Science, pp. 812–822, (1991).Google Scholar
  13. [GGST86]
    H. N. Gabow, Z. Galil, T. Spencer and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,” Combinatorica, 6 (2), pp. 109–122, (1986).Google Scholar
  14. [HR91a]
    T. S. Hsu and V. Ramachandran, “A linear time algorithm for triconnectivity augmentation,” 32nd Annual Symposium on Foundations of Computer Science, pp. 548–559, (1991).Google Scholar
  15. [HR91b]
    T. S. Hsu and V. Ramachandran, “On finding a smallest augmentation to biconnect a graph,” 2nd Annual International Symposium on Algorithms, Springer Verlag LNCS 557, pp. 326–335, (1991).Google Scholar
  16. [HT84]
    D. Harel and R. E. Tarjan, “Fast algorithms for finding nearest common ancestors,” SIAM Journal on Computing, Vol 13, No. 2, pp. 338–355, (1984).CrossRefGoogle Scholar
  17. [KB91]
    G. Kant and H. Bodlaender, “Planar Graph Augmentation Problems,” 1991 Workshop on Algorithms and Data Structures, pp. 286–298, (1991).Google Scholar
  18. [KT86]
    A.V. Karzanov and E.A, Timofeev, “Efficient algorithm for finding all minimal edge cuts of a nonoriented graph,” Cybernetics, pp. 156–162, Translated from Kibernetika, No. 2, pp. 8–12, (1986).Google Scholar
  19. [KV92]
    S. Khuller and U. Vishkin, “Biconnectivity approximations and graph carvings,” Technical Report UMIACS-TR-92-5, CS-TR-2825, Univ. of Maryland, January (1992), Also to appear in 24th Annual Symposium on Theory of Computing, (1992).Google Scholar
  20. [NGM90]
    D. Naor, D. Gusfield and C. Martel, “A fast algorithm for optimally increasing the edge-connectivity,” 31st Annual Symposium on Foundations of Computer Science, pp. 698–707, (1990).Google Scholar
  21. [RG77]
    A. Rosenthal and A. Goldner, “Smallest augmentations to biconnect a graph,” SIAM Journal on Computing, Vol. 6, No. 1, pp. 55–66, (1977).CrossRefGoogle Scholar
  22. [Wa88]
    T. Watanabe, “An efficient augmentation to k-edge connect a graph,” Tech. Report C-23, Department of Applied Math., Hiroshima University, April 1988.Google Scholar
  23. [WN87]
    T. Watanabe and A. Nakamura, “Edge-connectivity augmentation problems,” Journal of Comp. and Sys. Sciences, 35 (1), pp. 96–144, (1987).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Samir Khuller
    • 1
  • Ramakrishna Thurimella
    • 2
  1. 1.Institute for Advanced Computer Studies (UMIACS)The University of Maryland at College ParkCollege Park
  2. 2.Department of Mathematics and Computer ScienceUniversity of DenverDenver

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