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A Heuristic for the Stacker Crane Problem on Trees Which Is Almost Surely Exact

  • Amin Coja-Oghlan
  • Sven O. Krumke
  • Till Nierhoff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2906)

Abstract

Given an edge-weighted transportation network G and a list of transportation requests L, the stacker crane problem is to find a minimum-cost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper, we consider the case that the transportation network G is a tree, and that the requests are chosen randomly according to a certain class of probability distributions. We show that a polynomial time algorithm by Frederickson and Guan [11], which guarantees a 4/3-approximation in the worst case, on almost all inputs finds a minimum-cost tour, along with a certificate of the optimality of its output.

Keywords

Binary Tree Travelling Salesman Problem Chromatic Number Small Component Random Model 
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.

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References

  1. 1.
    Ascheuer, N., Krumke, S.O., Rambau, J.: Online dial-a-ride problems: Minimizing the completion time. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 639–650. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Atallah, M.J., Kosaraju, S.R.: Efficient solutions to some transportation problems with applications to minimizing robot arm travel. SIAM Journal on Computing 17, 849–869 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  4. 4.
    Ausiello, G., Feuerstein, E., Leonardi, S., Stougie, L., Talamo, M.: Algorithms for the on-line traveling salesman. Algorithmica 29, 560–581 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. In: Proc 35th SToC (2003)Google Scholar
  6. 6.
    Boppana, R.B.: Eigenvalues and graph bisection: An average case analysis. In: Proceedings of 28th FoCS (1987)Google Scholar
  7. 7.
    Burkard, R., Fruhwirth, B., Rote, G.: Vehicle routing in an automated warehouse: Analysis and optimization. Annals of Operations Research 57, 29–44 (1995)zbMATHCrossRefGoogle Scholar
  8. 8.
    Charikar, M., Raghavachari, B.: The finite capacity dial-A-ride problem. In: Proceedings of the 39th FoCS (1998)Google Scholar
  9. 9.
    Coja-Oglan, A., Krumke, S.O., Nierhoff, T.: Scheduling a server on a caterpillar network - a probabilistic analysis. In: Proceedings of the 6th Workshop on Models and Algorithms for Planning and Scheduling Problems (2003)Google Scholar
  10. 10.
    Feuerstein, E., Stougie, L.: On-line single server dial-a-ride problems. Theoretical Computer Science (to appear)Google Scholar
  11. 11.
    Frederickson, G.N., Guan, D.J.: Nonpreemptive ensemble motion planning on a tree. Journal of Algorithms 15, 29–60 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Frederickson, G.N., Hecht, M.S., Kim, C.E.: Approximation algorithms for some routing problems. SIAM Journal on Computing 7, 178–193 (1978)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Gröpl, C., Hougardy, S., Nierhoff, T., Prömel, H.J.: Approximation algorithms for the Steiner tree problem in graphs. In: Cheng, X., Du, D.Z. (eds.) Steiner Trees in Industry, pp. 235–279. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  14. 14.
    Guan, D.J.: Routing a vehicle of capacity greater than one. Discrete Applied Mathematics 81, 41–57 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hauptmeier, D., Krumke, S.O., Rambau, J., Wirth, H.C.: Euler is standing in line. Discrete Applied Mathematics 113, 87–107 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Janson, S., Luczak, T., Ruciński, A.: Random Graphs. John Wiley & Sons, Chichester (2000)zbMATHGoogle Scholar
  17. 17.
    Johnson, D.S., Papadimitriou, C.H.: Performance guarantees for heuristics. In: Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R., Shmoys, D.B. (eds.) The Travelling Salesman Problem, pp. 145–180. Wiley, Chichester (1985)Google Scholar
  18. 18.
    Knuth, D.E.: The art of computer programming. Sorting and searching, vol. 3. Addison-Wesley, Reading (1968)zbMATHGoogle Scholar
  19. 19.
    Kreuter, B., Nierhoff, T.: Greedily approximating the r-independent set and kcenter problems on random instances. In: Rolim, J.D.P. (ed.) RANDOM 1997. LNCS, vol. 1269, pp. 43–53. Springer, Heidelberg (1997)Google Scholar
  20. 20.
    Papadimitriou, C.H., Vempala, S.: On the approximability of the traveling salesman problem. In: Proc. of the 32nd SToC (2000)Google Scholar
  21. 21.
    Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Sven O. Krumke
    • 2
  • Till Nierhoff
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department OptimizationKonrad-Zuse-Zentrum für Informationstechnik BerlinBerlin-DahlemGermany

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