Advertisement

Hardness Results for the Probabilistic Traveling Salesman Problem with Deadlines

  • Dennis Weyland
  • Roberto Montemanni
  • Luca Maria Gambardella
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

The Probabilistic Traveling Salesman Problem with Deadlines (PTSPD) is a Stochastic Vehicle Routing Problem considering time dependencies. Even the evaluation of the objective function is considered to be a computationally demanding task. So far there is no evaluation method known that guarantees a polynomial runtime, but on the other hand there are also no hardness results regarding the PTSPD objective function. In our work we show that the evaluation of the objective function of the PTSPD, even for Euclidean instances, is #P-hard. In fact, we even show that computing the probabilities, with which deadlines are violated is #P-hard. Based on this result we additionally show that the decision variant of the Euclidean PTSPD, the optimization variant of the Euclidean PTSPD and delta evaluation in reasonable local search neighborhoods is #P-hard.

Keywords

stochastic combinatorial optimization stochastic vehicle routing computational complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baker, E.K.: An exact algorithm for the time-constrained traveling salesman problem. Operations Research 31(5), 938–945 (1983)zbMATHCrossRefGoogle Scholar
  2. 2.
    Balaprakash, P., Birattari, M., Stützle, T., Dorigo, M.: Estimation-based metaheuristics for the probabilistic traveling salesman problem. Computers & Operations Research 37(11), 1939–1951 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Birattari, M., Balaprakash, P., Stützle, T., Dorigo, M.: Estimation-based local search for stochastic combinatorial optimization using delta evaluations: A case study on the probabilistic traveling salesman problem. INFORMS Journal on Computing 20(4), 644–658 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Birge, J.R., Louveaux, F.: Introduction to stochastic programming. Springer (1997)Google Scholar
  5. 5.
    Campbell, A.M.: Aggregation for the probabilistic traveling salesman problem. Computers and Operations Research 33(9), 2703–2724 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Campbell, A.M., Thomas, B.W.: Probabilistic traveling salesman problem with deadlines. Transportation Science 42(1), 1–21 (2008)CrossRefGoogle Scholar
  7. 7.
    Campbell, A.M., Thomas, B.W.: Runtime reduction techniques for the probabilistic traveling salesman problem with deadlines. Computers and Operations Research 36(4), 1231–1248 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chepuri, K., Homem-de-Mello, T.: Solving the vehicle routing problem with stochastic demands using the cross-entropy method. Annals of Operations Research 134(1), 153–181 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Desrochers, M., Lenstra, J.K., Savelsbergh, M.W.P., Soumis, F.: Vehicle routing with time windows: optimization and approximation. In: Golden, B.L., Assad, A.A. (eds.) Vehicle Routing: Methods and Studies, pp. 65–84. Elsevier Science Publishers (1988)Google Scholar
  10. 10.
    Desrosiers, J., Sauvé, M., Soumis, F.: Lagrangian relaxation methods for solving the minimum fleet size multiple traveling salesman problem with time windows. Management Science 34(8), 1005–1022 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. Mathematical Programming 106(3), 423–432 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gendreau, M., Laporte, G., Seguin, R.: An exact algorithm for the vehicle routing problem with stochastic demands and customers. Transportation Science 29(2), 143 (1995)zbMATHCrossRefGoogle Scholar
  13. 13.
    Gendreau, M., Laporte, G., Seguin, R.: A tabu search heuristic for the vehicle routing problem with stochastic demands and customers. Operations Research, 469–477 (1996)Google Scholar
  14. 14.
    Gendreau, M., Laporte, G., Seguin, R.: Stochastic vehicle routing. European Journal of Operational Research 88(1), 3–12 (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    Jaillet, P.: Probabilistic traveling salesman problems. PhD thesis, M. I. T., Dept. of Civil Engineering (1985)Google Scholar
  16. 16.
    Johnson, D.S., McGeoch, L.A.: The traveling salesman problem: A case study in local optimization. Local Search in Combinatorial Optimization, 215–310 (1997)Google Scholar
  17. 17.
    Kolen, A.W.J., Rinnooy Kan, A.H.G., Trienekens, H.: Vehicle Routing with Time Windows. Operations Research 35(2), 266 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Laporte, G., Louveaux, F.V., Van Hamme, L.: An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Operations Research, 415–423 (2002)Google Scholar
  19. 19.
    Morris, B., Sinclair, A.: Random walks on truncated cubes and sampling 0-1 knapsack solutions. SIAM Journal on Computing 34, 195 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Salkin, H.M., De Kluyver, C.A.: The knapsack problem: a survey. Naval Research Logistics Quarterly 22(1), 127–144 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Savelsbergh, M.W.P.: Local search in routing problems with time windows. Annals of Operations Research 4(1), 285–305 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research 35(2), 254–265 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Weyland, D., Bianchi, L., Gambardella, L.M.: New Approximation-Based Local Search Algorithms for the Probabilistic Traveling Salesman Problem. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 681–688. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. 24.
    Weyland, D., Montemanni, R., Gambardella, L.M.: Heuristics for the probabilistic traveling salesman problem with deadlines using monte carlo sampling (2011) (submitted for publication)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dennis Weyland
    • 1
  • Roberto Montemanni
    • 1
  • Luca Maria Gambardella
    • 1
  1. 1.Istituto Dalle Molle di Studi sull’Intelligenza ArtificialeIDSIASwitzerland

Personalised recommendations