Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A Branch-and-Cut Algorithm for the Median-Path Problem


The Median-Path problem consists of locating a st-path on a network, minimizing a function of two parameters: accessibility to the path and total cost of the path. Applications of this problem can be found in transportation planning, water resource management and fluid transportation.

A problem formulation based on Subtour and Variable Upper Bound (VUB) inequalities was proposed in the seminal paper by (Current, Revelle and Cohon, 1989).

In this paper we introduce a tighter formulation, based on a new family of valid inequalities, named Lifted Subtour inequalities, that are proved to be facet-defining. For the class of Lifted Subtour inequalities we propose a polynomial separation algorithm. Then we introduce more families of valid inequalities derived by investigating the relation to the Asymmetric Traveling Salesman Problem (ATSP) polytope and to the Stable Set polytope.

These results are used to develop a Branch-and-Cut algorithm that enables us to solve to optimality small and medium size instances in less than 2 hours of CPU time on a workstation.

This is a preview of subscription content, log in to check access.


  1. 1.

    P. Avella and A. Sassano, “On the p-median polytope,” Mathematical Programming, vol. 89, pp. 395–411, 2001.

  2. 2.

    P. Avella, A. Sassano, and I. Vasil'ev, “Computational study of large-scale $p$-Median problems,” Technical Report 08-03, DIS – University of Rome “La Sapienza,” 2003.

  3. 3.

    P. Avella and A. Sforza, “Logical reduction tests for the p-Median problem,” Annals of Operations Research, vol. 86, pp. 105–115, 1999.

  4. 4.

    E. Balas and M. Fischetti, “A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets,” Mathematical Programming, vol. 58, pp. 325–352, 1993.

  5. 5.

    E. Balas and M. Oosten, “On the Cycle polytope of a directed graph,” Networks, vol. 36, no. 1, pp. 34–46, 2001.

  6. 6.

    G. Bruno, G. Ghiani, and G. Improta, “A multi-modal approach to the location of a rapid transit line,” European Journal of Operational Research, vol. 104, pp. 321–332, 1998.

  7. 7.

    F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, 1990.

  8. 8.

    J.R. Current, C.S. Revelle, and C.S. Cohon, “The Median Shortest Path Problem: A multiobjective approach to analyze cost vs. accessibility in the design of transportation networks,” Transportation Science, vol. 21, pp. 188–197, 1989.

  9. 9.

    R.W. Floyd, “Algorithm 97: Shortest path,” Comm. ACM, vol. 5, 1962.

  10. 10.

    M. Grotschel and M.W. Padberg, “Polyhedral theory,” in The Traveling Salesman Problem, Lenstra, Lawrel and Shmoys, Rinnooy Kan (Eds.), Wiley & Sons, 1985.

  11. 11.

    S.L. Hakimi, E.F. Schmeichel, and M. Labbé, “On locating path- or tree-shaped facilities on networks,” Networks, vol. 23, pp. 543–555, 1993.

  12. 12.

    M. Labbé, G. Laporte, and I. Rodriguez Martin, “Path, tree cycle location,” in Fleet Management and Logistics, T.G. Crainic and G. Laporte (Eds.), Kluwer, 1998.

  13. 13.

    M. Labbé, G. Laporte, I. Rodriguez Martin, and J.J. Salazar, “The median cycle problem,” Technical Report ULB-SMG-01, 2001.

  14. 14.

    E. Minieka, “The optimal location of a path or tree in a tree network,” Networks, vol. 15, pp. 309–321, 1985.

  15. 15.

    E. Minieka and N.H. Patel, “On finding the core of a tree with a specified length,” Journal of Algorithms, vol. 4, pp. 345–352, 1983.

  16. 16.

    C.A. Morgan and P.J. Slater, “A linear algorithm for a core of a tree,” Journal of Algorithms, vol. 1, pp. 247–258, 1980.

  17. 17.

    M.W. Padberg, “A note on zero-one programming,” Operations Research, vol. 23, pp. 833–837, 1975.

  18. 18.

    M. Queyranne and Y. Wang, “Hamiltonian path and symmetric traveling salesman polytopes,” Mathematical Programming, vol. 58, pp. 89–110, 1993.

  19. 19.

    M.B. Richey, “Optimal location of a path or tree on a network with cycles,” Networks, vol. 20, pp. 391–407, 1990.

  20. 20.

    P.J. Slater, “Locating central paths in a graph,” Transportation Science, vol. 16, pp. 1–18, 1982.

Download references

Author information

Correspondence to Pasquale Avella.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Avella, P., Boccia, M., Sforza, A. et al. A Branch-and-Cut Algorithm for the Median-Path Problem. Comput Optim Applic 32, 215–230 (2005).

Download citation


  • Path-Location
  • Median-Path
  • Branch-and-Cut