Stochastic Vehicle Routing with Recourse

  • Inge Li Gørtz
  • Viswanath Nagarajan
  • Rishi Saket
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed – but costs here become more expensive by a factor λ.

We present an O(log2 n ·log())-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based ω(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.

Keywords

Approximation Algorithm Vertex Cover Vehicle Route Problem Submodular Function Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ak, A., Erera, A.L.: A paired-vehicle recourse strategy for the vehicle-routing problem with stochastic demands. Transportation Science 41(2), 222–237 (2007)CrossRefGoogle Scholar
  2. 2.
    Altinkemer, K., Gavish, B.: Heuristics for unequal weight delivery problems with a fixed error guarantee. Operations Research Letters 6, 149–158 (1987)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bansal, N., Khot, S.: Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 250–261. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Bertsimas, D.J.: A vehicle routing problem with stochastic demand. Operations Research 40(3), 574–585 (1992)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bertsimas, D.J., Jaillet, P., Odoni, A.R.: A priori optimization. Operations Research 38(6), 1019–1033 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)MATHGoogle Scholar
  7. 7.
    Calinescu, G., Zelikovsky, A.: The polymatroid steiner problems. Journal of Combinatorial Optimization 9(3), 281–294 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Campbell, A.M., Thomas, B.W.: Probabilistic traveling salesman problem with deadlines. Transportation Science 42(1), 1–21 (2008)CrossRefGoogle Scholar
  9. 9.
    Charikar, M., Chekuri, C., Goel, A., Guha, S.: Rounding via trees: Deterministic approximation algorithms for group steiner trees and k-median. In: Proc. STOC 1988, pp. 114–123 (1998)Google Scholar
  10. 10.
    Charikar, M., Chekuri, C., Pál, M.: Sampling Bounds for Stochastic Optimization. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 257–269. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Chekuri, C., Even, G., Kortsarz, G.: A greedy approximation algorithm for the group steiner problem. Discrete Applied Mathematics 154(1), 15–34 (2006)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Chekuri, C., Pál, M.: A recursive greedy algorithm for walks in directed graphs. In: Proc. FOCS, pp. 245–253 (2005)Google Scholar
  13. 13.
    Dean, B.C., Goemans, M.X., Vondrák, J.: Approximating the stochastic knapsack problem: The benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dror, M.: Vehicle Routing with Stochastic Demands: Models & Computational Methods. In: Modeling Uncertainty. International Series In Operations Research & Management Science, vol. 46(8), pp. 625–649. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Erera, A.L., Savelsbergh, M.W.P., Uyar, E.: Fixed routes with backup vehicles for stochastic vehicle routing problems with time constraints. Networks 54(4), 270–283 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Garg, N., Konjevod, G., Ravi, R.: A Polylogarithmic Approximation Algorithm for the Group Steiner Tree Problem. Journal of Algorithms 37(1), 66–84 (2000)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Gørtz, I.L., Nagarajan, V., Saket, R.: Stochastic vehicle routing with recourse. CoRR, abs/1202.5797 (2012)Google Scholar
  18. 18.
    Gupta, A., Krishnaswamy, R., Nagarajan, V., Ravi, R.: Approximation Algorithms for Stochastic Orienteering. In: Proc. SODA 2012, pp. 245–253 (2012)Google Scholar
  19. 19.
    Gupta, A., Nagarajan, V., Ravi, R.: Approximation Algorithms for VRP with Stochastic Demands. Operations Research 60(1), 123–127 (2012)MATHCrossRefGoogle Scholar
  20. 20.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: Sampling and cost-sharing: Approximation algorithms for stochastic optimization problems. SIAM J. Comput. 40(5), 1361–1401 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Mathematics of Operations Research 10(4), 527–542 (1985)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Immorlica, N., Karger, D.R., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proc. SODA 2004, pp. 691–700 (2004)Google Scholar
  23. 23.
    Katriel, I., Mathieu, C.K., Upfal, E.: Commitment under uncertainty: Two-stage stochastic matching problems. Theor. Comput. Sci. 408(2-3), 213–223 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. STOC 2002, pp. 767–775 (2002)Google Scholar
  25. 25.
    Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Struct. Algorithms 20(3), 465–482 (2002)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. Math. Program. 108(1), 97–114 (2006)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Savelsbergh, M.W.P., Goetschalkx, M.: A comparison of the efficiency of fixed versus variable vehicle routes. J. Business Logistics 16, 163–187 (1995)Google Scholar
  29. 29.
    Schrijver, A.: Combinatorial optimization: polyhedra and efficiency. Springer, Berlin (2003)MATHGoogle Scholar
  30. 30.
    Shmoys, D.B., Talwar, K.: A Constant Approximation Algorithm for the a priori Traveling Salesman Problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 331–343. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  31. 31.
    Shmoys, D.B., Sozio, M.: Approximation Algorithms for 2-Stage Stochastic Scheduling Problems. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 145–157. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  32. 32.
    Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM 53(6), 978–1012 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Srinivasan, A.: New approaches to covering and packing problems. In: Proc. SODA 2001, pp. 567–576 (2001)Google Scholar
  34. 34.
    Stewart, W., Golden, B.: Stochastic vehicle routing: A comprehensive approach. Eur. Jour. Oper. Res. 14, 371–385 (1983)MATHCrossRefGoogle Scholar
  35. 35.
    Toth, P., Vigo, D.: The vehicle routing problem. Society for Industrial and Applied Mathematics (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Inge Li Gørtz
    • 1
  • Viswanath Nagarajan
    • 2
  • Rishi Saket
    • 2
  1. 1.DTU InformaticsTechnical University of DenmarkDenmark
  2. 2.IBM T.J. Watson Research CenterUSA

Personalised recommendations