Annals of Operations Research

, Volume 57, Issue 1, pp 233–249 | Cite as

On the core of the minimum cost steiner tree game in networks

  • Darko Skorin-Kapov
Article

Abstract

A cost allocation problem arising from the Steiner Tree (ST) problem in networks is analyzed. This cost allocation problem is formulated as a cost cooperative game in characteristic function form, referred to as theST-game. The class ofST games generalizes the class of minimum cost spanning tree games which were used in the literature to analyze a variety of cost allocation problems. In general, the core of anST-game may be empty. We construct an efficient Core Heuristic to compute a “good” lower bound on the maximum fraction of the total cost that can be distributed among users while satisfying the core constraints. Based on the Core Heuristic, we also provide a sufficient condition for a givenST not to be optimal for the linear programming relaxation of an integer programming formulation of theST problem. The Core Heuristic was implemented and tested on 76 data sets from the literature (Wong's, Aneja's and Beasley's Steiner tree problems). Core points were found for 69 of these cases, and points “close” to the core were computed in the others.

Keywords

Steiner tree game theory cost allocation integer programming linear programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Y.P. Aneja, An integer linear programming approach to the Steiner problem in graphs, Networks 10(1989)167–178.Google Scholar
  2. [2]
    J.E. Beasley, An SST-based algorithm for the Steiner problem in graphs, Networks 19(1989)1–16.Google Scholar
  3. [3]
    C. Bird, On cost allocation for a spanning tree: A game theoretic approach, Networks 6(1976)335–350.Google Scholar
  4. [4]
    S. Chopra, E.R. Gorres and M.R. Rao, Solving the Steiner tree problem on a graph using branch and cut, ORSA J. Comp. 4(1992)320–335.Google Scholar
  5. [5]
    M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, CA, 1979).Google Scholar
  6. [6]
    D. Granot, A generalized linear production model: A unifying model, Math. Progr. 34(1986)212–223.Google Scholar
  7. [7]
    D. Granot and F. Granot, A fixed-cost spanning-forest problem, Math. Oper. Res. 17(1992)765–780.Google Scholar
  8. [8]
    D. Granot and G. Huberman, Minimum cost spanning tree games, Math. Progr. 21(1981)1–18.Google Scholar
  9. [9]
    D. Granot and G. Huberman, On the core and nucleolus of M.C.S.T. games, Math. Progr. 29(1984)323–347.Google Scholar
  10. [10]
    M. Grötschel, L. Lovász and A. Schrijver,Geometric Algorithms and Combinatorial Optimization (Springer, Berlin/Heidelberg, 1988).Google Scholar
  11. [11]
    E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).Google Scholar
  12. [12]
    S.C. Littlechild, A simple expression for the nucleolus in a special case, Int. J. Game Theory 3(1974)21–29.Google Scholar
  13. [13]
    N. Megiddo, Computational complexity and the game theory approach to cost allocation for a tree, Math. Oper. Res. 3(1978)189–196.Google Scholar
  14. [14]
    A. Prodon, M. Liebling and H. Gröflin, Steiner's problem on two-trees, Working Paper RO 850315, Department of Mathematics, EPF Lausanne, Switzerland (March 1985).Google Scholar
  15. [15]
    W.W. Sharkey, Cores of games with fixed costs and shared facilities, Int. Econ. Rev. 30(1990)245–262.Google Scholar
  16. [16]
    W.W. Sharkey, Cost allocation theory applied to the telecommunications network, Working Paper, Bellcore (May 1992).Google Scholar
  17. [17]
    A. Tamir, On the core of network synthesis game, Math. Progr. 50(1991)123–135.Google Scholar
  18. [18]
    R.T. Wong, A dual ascent approach for Steiner tree problems on a directed graph, Math. Progr. 28(1984)271–287.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Darko Skorin-Kapov
    • 1
  1. 1.Harriman School for Management and PolicyState University of New York at Stony BrookUSA

Personalised recommendations