Advertisement

Polynomial Time Algorithms for Edge-Connectivity Augmentation of Hamiltonian Paths

  • Anna Galluccio
  • Guido Proietti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2223)

Abstract

Given a graph G of n vertices and m edges, and given a spanning subgraph H of G, the problem of finding a minimum weight set of edges of G, denoted as Aug2(H,G), to be added to H to make it 2-edge connected, is known to be NP-hard. In this paper, we present polynomial time effcient algorithms for solving the special case of this classic augmentation problem in which the subgraph H is a Hamiltonian path of G. More precisely, we show that if G is unweighted, then Aug2(H,G) can be computed in O(m) time and space, while if G is non-negatively weighted, then Aug2(H,G) can be comp uted in O(m+n log n) time and O(m) space. These results have an interesting application for solving a survivability problem on communication networks.

Keywords

Graph Hamiltonian path augmentation 2-edge connectivity network survivability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. V. Aho, J. E. Hopcroft and J. D. Ullman, The design and analysis of computeralgorithms, Addison Wesley, (1974).Google Scholar
  2. 2.
    G. S. Brodal, Worst-case effcient priority queues, Proc. 7th Annual ACM-SIAMSymp. on Discrete Algorithms (SODA’96), ACM/IEEE Computer Society, 52–58.Google Scholar
  3. 3.
    K. P. Eswaran and R. E. Tarjan, Augmentation problems, SIAM Journal on Computing,5 (1976) 653–665.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM Journalon Discrete Mathematics, 5 (1992) 25–53.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. N. Frederickson and J. Jájá, On the relationshipb etween the biconnectivityaugmentation problems, SIAM Journal on Computing, 10 (1981) 270–283.MathSciNetCrossRefGoogle Scholar
  6. 6.
    H. N. Gabow, Application of a poset representation to edge connectivity andgraph rigidity, Proc. 32nd Ann. IEEE Symp. on Foundations of Computer Science(FOCS’91), IEEE Computer Society, 812–821.Google Scholar
  7. 7.
    A. Galluccio and G. Proietti, Towards a 4/3-approximation algorithm for biconnectivity,IASI-CNR Technical report R. 506, June 1999. Submitted for publication.Google Scholar
  8. 8.
    M. Grötschel, C. L. Monma and M. Stoer, Design of survivable networks, in Handbooks in OR and MS, Vol. 7, Elsevier (1995) 617–672.MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Khuller and R. Thurimella, Approximations algorithms for graph augmentation,Journal of Algorithms, 14 (1993) 214–225.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Khuller, Approximation algorithms for finding highly connected subgraphs, in Approximation Algorithms for NP-Hard Problems, Dorit S. Hochbaum Eds., PWSPublishing Company, Boston, MA, 1996.Google Scholar
  11. 11.
    S. Khuller and U. Vishkin, Biconnectivity approximations and graph carvings,Journal of the ACM, 41(2) (1994) 214–235.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. H. Möhring, F. Wagner and D. Wagner, VLSI network design, in Handbooks inOR and MS, Vol. 8, Elsevier (1995) 625–712.MathSciNetzbMATHGoogle Scholar
  13. 13.
    H. Nagamochi and T. Ibaraki, An approximation for finding a smallest 2-edgeconnected subgraph containing a specified spanning tree, Proc. 5th Annual InternationalComputing and Combinatorics Conference (COCOON’99), Vol. 1627 of Lecture Notes in Computer Science, Springer, 31–40.zbMATHGoogle Scholar
  14. 14.
    S. Vempala and A. Vetta, Factor 4/3 approximations for minimum 2-connectedsubgraphs, Proc. 3rd Int. Workshop on Approximation Algorithms for CombinatorialOptimization Problems (APPROX 2000), Vol. 1913 of Lecture Notes in ComputerScience, Springer, 262–273.Google Scholar
  15. 15.
    T. Watanabe and A. Nakamura, Edge-connectivity augmentation problems, Journalof Computer and System Sciences, 35(1) (1987) 96–144.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    P. Winter, Steiner problem in networks: a survey, Networks, 17 (1987) 129–167.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Anna Galluccio
    • 1
  • Guido Proietti
    • 2
  1. 1.Istituto di Analisi dei Sistemi ed InformaticaRomaItaly
  2. 2.Dipartimento di Matematica Pura ed ApplicataUniversitá dell’AquilaL’AquilaItaly

Personalised recommendations