Reverse Center Location Problem

  • Jianzhong Zhang
  • Xiaoguang Yang
  • Mao-cheng Cai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)


In this paper we consider a reverse center location problem in which we wish to spend as less cost as possible to ensure that the di- stances from a given vertex to all other vertices in a network are within given upper bounds. We first show that this problem is NP-hard. We then formulate the problem as a mixed integer programming problem and propose a heuristic method to solve this problem approximately on a spanning tree. A strongly polynomial method is proposed to solve the reverse center location problem on this spanning tree.


networks and graphs NP-hard satisfiability problem relaxation maximum cost circulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burton, D., Pulleyblank, W.R., Toint, Ph.L.: The inverse shortest paths with upper bounds on shortest path costs. Report 93/03, (1993), Department of Mathematics, Facultes, Univeristaires ND de la Paix, B-5000 Nemur, BelgiumGoogle Scholar
  2. 2.
    Burton, D., Toint, Ph.L.: On an instance of the inverse shortest paths problem. Mathematical Programming 53 (1992) 45–61Google Scholar
  3. 3.
    Cai, M., Li, Y.: Inverse matroid intersection problem. ZOR Mathematical Methods of Operations Research 45 (1997) 235–243Google Scholar
  4. 4.
    Cai, M., Yang, X., Zhang, J.: The complexity analysis of the inverse center location problem. Journal of Global Optimization (to appear)Google Scholar
  5. 5.
    Christofides, N.: Graph Theory: An Algorithmic Approach. Academic Press Inc. (London) Ltd, (1975)Google Scholar
  6. 6.
    Cook, S.A.: The complexity of theorem proving procedures. Proc. 3rd ACM Symp. on the Theory of Computing, ACM(1971), 151–158Google Scholar
  7. 7.
    Fekete, S., Kromberg, S., Hochstattler, W., Moll, C.: The Complexity of an Inverse Shortest Paths Problem. Working paper (1993), University of Cologne.Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide of the Theory of NP-Completeness. Freeman, San Francisco, (1979)Google Scholar
  9. 9.
    Grötschel, M., Jünger, M., Reinelt, G.: On the acyclic subgraph polytope. Mathematical Programming 33 (1985) 28–42Google Scholar
  10. 10.
    Hochbaum, D.(eds.): Approximation Algorithms for NP-hard Problems. PWS, Boston, 1997Google Scholar
  11. 11.
    Hu, Z., Liu, Z.: A strongly polynomial algorithm for the inverse shortest arborescence problem. Discrete Applied Mathematics 82 (1998) 135–154Google Scholar
  12. 12.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, (1976)Google Scholar
  13. 13.
    Leung, L.: A graph-theoretic heuristic for designing loop-layout manufacturing systems. European Journal of Operational Research, 57 (1992) 243–252Google Scholar
  14. 14.
    Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Proc. 20th ACM Symp. on the Theory of Comp., (1988), 377–387Google Scholar
  15. 15.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall Inc. Englewood Cliffs, New Jersey, (1982)Google Scholar
  16. 16.
    Tardos, É.: A strongly polynomial minimum cost circulation algorithm. Combinatorica 5 (1985) 247–255Google Scholar
  17. 17.
    Yang, C., Zhang, J.: Inverse maximum flow and minimum cut problems. Optimization 40 (1997) 147–170Google Scholar
  18. 18.
    Yang, C., Zhang, J.: Inverse maximum capacity problem. Operations Research Spektrum (to appear)Google Scholar
  19. 19.
    Yang, C., Zhang, J.: Two general methods for inverse optimization problems. Applied Mathematics Letters (to appear)Google Scholar
  20. 20.
    Zhang, J., Cai, M.: Inverse problem of minimum cuts. ZOR Mathematical Methods of Operations Research 48 (1998) 51–58Google Scholar
  21. 21.
    Zhang, J., Liu, Z.: Calculating some inverse linear programming problem. Journal of Computational and Applied Mathematics 72 (1996) 261–273Google Scholar
  22. 22.
    Zhang, J., Liu, Z., Ma, Z.: On inverse problem of minimum spanning tree with partition constraints. ZOR Mathematical Methods of Operations Research 44 (1996) 347–358Google Scholar
  23. 23.
    Zhang, J., Liu, Z., Ma, Z.: Inverse Fractional Matching Problem. The Journal of Australia Mathematics Society, Ser. B: Applied Mathematics (to appear)Google Scholar
  24. 24.
    Zhang, J., Ma, Z.: A network flow method for solving some inverse combinatorial optimization problems. Optimization 37 (1996) 59–72Google Scholar
  25. 25.
    Zhang, J., Ma, Z.: Solution structure of some inverse optimization problems. Journal of Combinatorial Optimization (to appear)Google Scholar
  26. 26.
    Zhang, J., Ma, Z., Yang, C.: A column generation method for inverse shortest path problem. ZOR Mathematical Methods of Operations Research 41 (1995) 347–358Google Scholar
  27. 27.
    Zhang, J., Xu, S., Ma, Z.: An algorithm for inverse minimum spanning tree problem. Optimization Methods and Software 8 (1997) 69–84Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jianzhong Zhang
    • 1
  • Xiaoguang Yang
    • 2
  • Mao-cheng Cai
    • 3
  1. 1.Department of MathematicsCity University of Hong KongHong Kong
  2. 2.Academia SinicaInstitute of Systems ScienceBeijingChina
  3. 3.Academia SinicaInstitute of Systems ScienceBeijingChina

Personalised recommendations