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.
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References
Baker, E.K.: An exact algorithm for the time-constrained traveling salesman problem. Operations Research 31(5), 938–945 (1983)
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)
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)
Birge, J.R., Louveaux, F.: Introduction to stochastic programming. Springer (1997)
Campbell, A.M.: Aggregation for the probabilistic traveling salesman problem. Computers and Operations Research 33(9), 2703–2724 (2006)
Campbell, A.M., Thomas, B.W.: Probabilistic traveling salesman problem with deadlines. Transportation Science 42(1), 1–21 (2008)
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)
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)
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)
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)
Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. Mathematical Programming 106(3), 423–432 (2006)
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)
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)
Gendreau, M., Laporte, G., Seguin, R.: Stochastic vehicle routing. European Journal of Operational Research 88(1), 3–12 (1996)
Jaillet, P.: Probabilistic traveling salesman problems. PhD thesis, M. I. T., Dept. of Civil Engineering (1985)
Johnson, D.S., McGeoch, L.A.: The traveling salesman problem: A case study in local optimization. Local Search in Combinatorial Optimization, 215–310 (1997)
Kolen, A.W.J., Rinnooy Kan, A.H.G., Trienekens, H.: Vehicle Routing with Time Windows. Operations Research 35(2), 266 (1987)
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)
Morris, B., Sinclair, A.: Random walks on truncated cubes and sampling 0-1 knapsack solutions. SIAM Journal on Computing 34, 195 (2004)
Salkin, H.M., De Kluyver, C.A.: The knapsack problem: a survey. Naval Research Logistics Quarterly 22(1), 127–144 (1975)
Savelsbergh, M.W.P.: Local search in routing problems with time windows. Annals of Operations Research 4(1), 285–305 (1985)
Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research 35(2), 254–265 (1987)
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)
Weyland, D., Montemanni, R., Gambardella, L.M.: Heuristics for the probabilistic traveling salesman problem with deadlines using monte carlo sampling (2011) (submitted for publication)
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Weyland, D., Montemanni, R., Gambardella, L.M. (2012). Hardness Results for the Probabilistic Traveling Salesman Problem with Deadlines. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_35
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DOI: https://doi.org/10.1007/978-3-642-32147-4_35
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