Journal of Scheduling

, Volume 8, Issue 3, pp 197–210 | Cite as

A Hard Dial-a-Ride Problem that is Easy on Average

  • Amin Coja-Oghlan
  • Sven O. KrumkeEmail author
  • Till Nierhoff


In the dial-a-ride-problem (Darp) objects have to be moved between given sources and destinations in a transportation network by means of a server. The goal is to find the shortest transportation for the server. We study the Darp when the underlying transportation network forms a caterpillar. This special case is strongly NP-hard in the worst case. We prove that in a probabilistic setting there exists a polynomial time algorithm that finds an optimal solution with high probability. Moreover, with high probability the optimality of the solution found can be certified efficiently. In addition, we examine the complexity of the Darp in a semirandom setting and in the unweighted case.


average-case analysis NP-hard approximation algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alimonti, P. and V. Kann, “Hardness of approximating problems on cubic graphs,” in Proceedings of the 3rd Italian Conference on Algorithms and Complexity, vol. 1203 of Lecture Notes in Computer Science, Springer, 1997, pp. 288–298.Google Scholar
  2. Ascheuer, N., S. O. Krumke, and J. Rambau, “Online dial-a-ride problems: Minimizing the comple tion time,” in Proceedings of the 17th International Symposium on Theoretical Aspects of Computer Science, vol. 1770 of Lecture Notes in Computer Science, Springer, 2000, pp. 639–650.Google Scholar
  3. Atallah M. J. and S. R. Kosaraju, “Efficient solutions to some transportation problems with applications to minimizing robot arm travel,” SIAM lournal on Computing, 17, 849–869 (1988).CrossRefGoogle Scholar
  4. Ausiello, G., E. Feuerstein, S. Leonard, L. Stougie, and M. Talamo, “Algorithms for the on-line traveling salesman,” Algorithmica, 29, 560–581 (2001).CrossRefGoogle Scholar
  5. Bern, M. and P. Plassmann, “The Steiner problem with edge lengths 1 and 2,” Information Processing Letters, 32, 171–176 (1989).CrossRefGoogle Scholar
  6. Bollobás, B. Random graphs, Cambridge University Press, Cambridge, UK, 2 ed., 2001.Google Scholar
  7. Borodin, A. and R. El-Yaniv, Online Computation and Competitive Analysis, Cambridge University Press, 1998.Google Scholar
  8. Coja-Oghlan, A., S. O. Krumke, and T. Nierhoff, “A heuristic for the stacker crane problem on trees which is almost surely exact,” Journal of Algorithms, to appear.Google Scholar
  9. Feige, U. and I. Kilian, “Heuristics for semirandom graph problems,” Journal of Computer and System Sciences, 63, 639–671 (2001).CrossRefGoogle Scholar
  10. Feller, W., An Introduction to Probability Theory and Its Applications, vol. 1, John Wiley & Sons, Inc., 3 ed. 1968.Google Scholar
  11. Feuerstein, E. and L. Stougie, “On-line single server dial-a-ride problems,” Theoretical Computer Science, 268, 91–105 (2001).CrossRefGoogle Scholar
  12. Frederickson, G. N. “A note on the complexity of a simple transportation problem,” SIAM Journal on Com puting, 22, 57–61 (1993).CrossRefGoogle Scholar
  13. Frederickson, G. N. and D. J. Guan, “Nonpreemptive ensemble motion planning on a tree,” Journal of Algorithms, 15, 29–60 (1993).CrossRefGoogle Scholar
  14. Frederickson, G. N., M. S. Hecht, and C. E. Kim, “Approximation algorithms for some routing problems,” SIAM Journal on Computing, 7, 178–193 (1978).CrossRefGoogle Scholar
  15. Frieze, A. and C. McDiarmid, “Algorithmic theory of random graphs,” Random Structures and Algorithms, 10, 5–42 (1997).CrossRefGoogle Scholar
  16. Grimmett, G. R. and D. R. Stirzaker, Probability and Random Processes, Oxford Science Publications, 1988.Google Scholar
  17. Grötschel, M., S. O. Krumke, and J. Rmbau (eds)., Online Optimization of Large Scale Systems, Springer, Berlin Heidelberg New York, 2001.Google Scholar
  18. Hauptmeier, D., S. O. Krumke, and J. Rambau, “The online dial-a-ride problem under reasonable load,” Theoretical Computer Science. To appear. A preliminary version appeared in the Proceedings of the 4th Italian Conference on Algorithms and Complexity, vol. 1767 of Lecture Notes in Computer Science, 2000.Google Scholar
  19. Hauptmeier, D., S. O. Krumke, J. Rambau, and H.-C. Wirth, “Euler is standing in line,” Discrete Applied Mathematics, 113, 87–107 (2001). A preliminary version appeared in the Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science, vol. 1665 of Lecture Notes in Computer Science, 2000.Google Scholar
  20. Janson, S., T. Mluczak, and A. Ruciński, Random Graphs, John Wiley, 2000.Google Scholar
  21. Vazirani, V., Approximation Algorithms, Springer, 2001.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Sven O. Krumke
    • 2
    Email author
  • Till Nierhoff
    • 1
  1. 1.Institut für Informatik Unter den Linden 6Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations