Journal of Global Optimization

, Volume 25, Issue 3, pp 243–261 | Cite as

Computation of the Reverse Shortest-Path Problem

  • Jianzhong Zhang
  • Yixun Lin


The shortest-path problem in a network is to find shortest paths between some specified sources and terminals when the lengths of edges are given. This paper studies a reverse problem: how to shorten the lengths of edges with as less cost as possible such that the distances between specified sources and terminals are reduced to the required bounds. This can be regarded as a routing speed-up model in transportation networks. In this paper, for the general problem, the NP-completeness is shown, and for the case of trees and the case of single source-terminal, polynomial-time algorithms are presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bellman, R. (1958), On a routing problem, Quar. Appl. Math. 16, 87-90.Google Scholar
  2. 2.
    Berman, O., Ingco, D.I. and Odoni, A.R. (1992), Improving the location of minisum facilities through network modification, Annals of Oper. Res. 40, 1-16.Google Scholar
  3. 3.
    Burton, D. and Toint, Ph.L. (1992), On an instance of the inverse shortest paths problem, Math. Program. 53, 45-61.Google Scholar
  4. 4.
    Burton, D. and Toint, Ph.L. (1994), On the use of an inverse shortest paths algorithm for recovering linearly correlated costs, Math. Program. 63, 1-22.Google Scholar
  5. 5.
    Burton, D., Pulleyblank, W.R. and Toint, Ph.L. (1997), The inverse shortest paths problem with upper bounds on shortest paths costs, Lecture Notes in Econom. and Math. System 450, 156-171.Google Scholar
  6. 6.
    Cai, M., Yang, X. and Zhang, J. (1999), The complexity analysis of the inverse center location problem, J. Global Optimization 15, 213-218.Google Scholar
  7. 7.
    Drangmeister, K.U., Krumke, S.O., Marathe, M.V., Noltemeier, H. and Ravi, S.S. (1998), Modifying edges of a network to obtain short subgraphs, Theoretical Computer Science 203, 91-121.Google Scholar
  8. 8.
    Dreyfus, S.E. and Law, A.M. (1977), The Art and Theory of Dynamic Programming, Academic Press, New York.Google Scholar
  9. 9.
    Fulkerson, D.R. and Harding, G.C. (1977), Maximizing the minimum source-sink path subject to a budget constraint, Math. Program. 13, 116-118.Google Scholar
  10. 10.
    Garey, M.R. and Johnson, D.S. (1979), Computers and Intractability: A Guide to the theory of NP-Completeness, Freeman, San Francisco, CA.Google Scholar
  11. 11.
    Lawler, E.L. (1976), Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York.Google Scholar
  12. 12.
    Papadimitriou, C.H. and Steiglitz, K. (1982), Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  13. 13.
    Tarjan, R.E. (1985), Shortest path algorithms, Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), Wiley, New York, 753-759.Google Scholar
  14. 14.
    Zhang, J., Ma, Z. and Yang, C. (1995), A column generation method for inverse shortest path problem, ZOR Math. Methods of Oper. Res. 41, 347-358.Google Scholar
  15. 15.
    Zhang, J. and Ma, Z. (1996), A network flow method for solving some inverse combinatorial optimization problems, Optimization 37, 59-72.Google Scholar
  16. 16.
    Zhang, J., Liu, Z. and Ma, Z. (2000), Some reverse location problem, European J. Oper. Res. 124, 77-88.Google Scholar
  17. 17.
    Zhang, J., Yang, X. and Cai, M. (1999), Reverse center location problem, Lecture Notes in Computer Science 1741, 279-294.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Jianzhong Zhang
    • 1
  • Yixun Lin
    • 2
  1. 1.Department of MathematicsCity University of Hong KongHong Kong
  2. 2.Department of MathematicsZhengzhou UniversityZhengzhouChina

Personalised recommendations