The Uncapacitated Asymmetric Traveling Salesman Problem with Multiple Stacks

  • Sylvie Borne
  • Roland Grappe
  • Mathieu Lacroix
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)


In the uncapacitated asymmetric traveling salesman problem with multiple stacks, one first performs a hamiltonian circuit to pick up n items, storing them in a vehicle with k stacks satisfying last-in-first-out constraints, and then delivers every item by performing a second hamiltonian circuit. Here, we are interested in the convex hull of the arc-incidence vectors of such couples of hamiltonian circuits.

For the general case, we determine the dimension of this polytope, and show that every facet of the asymmetric traveling salesman polytope defines one of its facets. For the special case with two stacks, we provide an integer linear programming formulation whose linear relaxation is polynomial-time solvable, and we propose new families of valid inequalities to reinforce the latter.


asymmetric traveling salesman problem stacks polytope facets formulation valid inequalities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alba, M., Cordeau, J.-F., Dell’Amico, M., Iori, M.: A Branch-and-Cut Algorithm for the Double Traveling Salesman Problem with Multiple Stacks. INFORMS Journal on Computing (2011) (published online)Google Scholar
  2. 2.
    Bonomo, F., Mattia, S., Oriolo, G.: Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem. Theoretical Computer Science 412(45), 6261–6268 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carrabs, F., Cerulli, R., Speranza, M.G.: A Branch-and-Bound Algorithm for the Double TSP with Two Stacks. Technical report (2010)Google Scholar
  4. 4.
    Casazza, M., Ceselli, A., Nunkesser, M.: Efficient algorithms for the double traveling salesman problem with multiple stacks. Computers & Operations Research 39, 1044–1053 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Felipe, A., Ortuno, M.T., Tirado, G.: The Double Traveling Salesman Problem with Multiple Stacks: A Variable Neighborhood Search Approach. Computers & Operations Research 36, 2983–2993 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gutan, G., Punnen, A.P.: The Traveling Salesman Problem and Its Variations. In: Combinatorial Optimization, vol. 12. Kluwer Academic Publishers (2002)Google Scholar
  7. 7.
    Lusby, R.M., Larsen, J., Ehrgott, M., Ryan, D.: An exact method for the double TSP with multiple stacks. International Transactions on Operations Research 17, 637–652 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Petersen, H.L., Archetti, C., Sperenza, M.G.: Exact Solutions to the Double Travelling Salesman Problem with Multiple Stacks. Networks 56(4), 229–243 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Petersen, H.L., Madsen, O.B.G.: The double travelling salesman problem with multiple stacks - Formulation and heuristic solution approaches. European Journal of Operation Research 198(1), 139–147 (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Pnueli, A., Lempel, A., Even, S.: Transitive orientation of graphs and identification of permutation graphs. Canadian Journal of Mathematics 23, 160–175 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Toulouse, S.: Approximability of the Multiple Stack TSP. In: International Symposium on Combinatorial Optimization (ISCO 2010), ENDM 813-820 (2010)Google Scholar
  12. 12.
    Toulouse, S., Wolfler Calvo, R.: On the Complexity of the Multiple Stack TSP, kSTSP. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 360–369. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sylvie Borne
    • 1
  • Roland Grappe
    • 1
  • Mathieu Lacroix
    • 1
  1. 1.Laboratoire d’Informatique de Paris-Nord, UMR CNRS 7030Université ParisVilletaneuseFrance

Personalised recommendations