Preprocessing the Steiner Problem in Graphs

  • Cees Duin
Chapter
Part of the Combinatorial Optimization book series (COOP, volume 6)

Abstract

For combinatorial optimization problems that are NP-hard, it is important, before running a time consuming algorithm, to try to reduce the input size of the problem. This is the objective of a so-called preprocessing algorithm. A renowned NP-hard problem is the Steiner Problem in Graphs (SPG). It considers a weighted graph, denoted here as G = (V,K,E,c) with V the set of vertices, K a subset of so-called special vertices, E the set of undirected edges, and c: E→ Z + a positive integer weight function on E. The problem is to find a tree S of minimum total edge weight that spans the vertices of K.

Keywords

Minimum Span Tree Steiner Tree Steiner Tree Problem Special Distance Special Vertex 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Balakrishnan and N.R. Patel, Problem Reduction Methods and a Tree Generation Algorithm for the Steiner Network Problem, Networks 17 (1987) pp. 65–85.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J.E. Beasley, An Algorithm for the Steiner problem in Graphs, Networks 14 (1984) pp. 147–159.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J.E. Beasley, An SST based algorithm for the Steiner problem in graphs, Networks 19 (1989) pp. 1–16.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    S. Chopra, M.R. Rao and E.R. Gorres, Solving the Steiner Tree Problem on a Graph Using Branch and Cut, ORSA Journal on Computing 4 (1992), pp.320–335.MATHCrossRefGoogle Scholar
  5. [5]
    E.W. Dijkstra, A Note on Two Problems in Connexion with Graphs, Numerische Mathematik 1 (1959), pp. 269–271.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    C.W. Duin and A. Volgenant, An Edge Elimination Test for the Steiner Problem in Graphs, Operations Research Letters 8 (1989), pp. 79–83.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    C.W. Duin and A. Volgenant, Reduction Tests for the Steiner Problem in Graphs, Networks 19 (1989), pp. 549–567.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    C.W. Duin, Steiner’s Problem in Graphs, PhD Thesis University of Amsterdam (1993).Google Scholar
  9. [9]
    C.W. Duin, Reducing the Graphical Steiner Problem with a Sensitivity Test, to appear in Proceedings of DIMACS workshop on Network Design (1997)Google Scholar
  10. [10]
    Duin, C.W. and S. Voss, Efficient Path and Vertex Exchange in Steiner Tree Algorithms, Networks 29 (1997), pp. 89–105.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Dreyfus, S.E. and R.A. Wagner, The Steiner Problem in Graphs, Networks 1 (1972) pp. 195–207.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    M.L. Fredman and R.E. Tarjan, Fibonacci Heaps and Their Uses in improved Network Optimization Algorithms, Journal of the ACM 6 (1987), pp. 596–615MathSciNetCrossRefGoogle Scholar
  13. [13]
    Gomory, R.E. and T.C. Hu (1961), Multi-Terminal Network Flows, Journal of SIAM 9, pp. 551–556.MathSciNetMATHGoogle Scholar
  14. [14]
    Hwang, F.K., D.S. Richards and P. Winter, The Steiner Tree Problem, Annals of Discrete Mathematics 53 (1992).Google Scholar
  15. [15]
    T. Koch and A. Martin, Solving Steiner Tree Problems in Graphs to Optimality, to appear in Networks Google Scholar
  16. [16]
    K. Mehlhorn, A Faster Approximation Algorithm for the Steiner Problem in Graphs, Information Processing Letters 27 (1988), pp.125–128.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Polzin, T. and Daneshmand S.V., Improved algorithms for the Steiner Problem in Networks, Technical Report 06/1998, Theoretische Informatik Universität Mannheim (1998).Google Scholar
  18. [18]
    P.M. Spira, On Finding and Updating Spanning Trees and Shortest Paths SIAM Journal on Computing 4 (1975), pp. 375–380.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    P Winter, The Steiner Problem in Networks: A Survey, Networks 17 (1987), pp. 185–212.CrossRefGoogle Scholar
  20. [20]
    P Winter, Reductions for the Rectilinear Steiner tree Problem, Networks 26 (1987), pp. 187–198.CrossRefGoogle Scholar
  21. [21]
    Wong, R.T., A dual ascent based approach for the Steiner tree problem in a directed graph, Mathematical Programming 28 (1984), pp. 271–287.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Cees Duin
    • 1
  1. 1.AKE, Faculteit Economische Wetenschappen en EconometrieUniversiteit van AmsterdamAmsterdamNetherlands

Personalised recommendations