Pareto Shortest Paths is Often Feasible in Practice

  • Matthias Müller-Hannemann
  • Karsten Weihe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2141)


We study the problem of finding all Pareto-optimal solutions for the multi-criteria single-source shortest-path problem with nonnegative edge lengths. The standard approaches are generalizations of label-setting (Dijkstra) and label-correcting algorithms, in which the distance labels are multi-dimensional and more than one distance label is maintained for each node. The crucial parameter for the run time and space consumption is the total number of Pareto optima. In general, this value can be exponentially large in the input size. However, in various practical applications one can observe that the input data has certain characteristics, which may lead to a much smaller number — small enough to make the problem efficiently tractable from a practical viewpoint.

In this paper, we identify certain key characteristics, which occur in various applications. These key characteristics are evaluated on a concrete application scenario (computing the set of best train connections in view of travel time, fare, and number of train changes) and on a simplified randomized model, in which these characteristics occur in a very purist form. In the applied scenario, it will turn out that the number of Pareto optima on each visited node is restricted by a small constant. To counter-check the conjecture that these characteristics are the cause of these uniformly positive results, we will also report negative results from another application, in which these characteristics do not occur.


Multi-criteria optimization Pareto set shortest paths railway networks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Brumbaugh-Smith and D. Shier. An empirical investigation of some bicriterion shortest path algorithms. European Journal of Operations Research, 43:216–224, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Ehrgott and X. Gandibleux. An annotated biliography of multiobjective combinatorial optimization. OR Spektrum, pages 425–460, 2000.Google Scholar
  3. 3.
    P. Hansen. Bicriteria path problems. In G. Fandel and T. Gal, editors, Multiple Criteria Decision Making Theory and Applications, volume 177 of Lecture Notes in Economics and Mathematical Systems, pages 109–127. Springer Verlag, Berlin, 1979.Google Scholar
  4. 4.
    O. Jahn, R. H. Möhring, and A. S. Schulz. Optimal routing of traffic flows with length restrictions. In K. Inderfurth et al., editor, Operations Research Proceedings 1999, pages 437–442. Springer, 2000.Google Scholar
  5. 5.
    K. Mehlhorn and G. Schäfer. A heuristic for Dijkstra’s algorithm with many targets and its use in weighted matching algorithms. In Proceedings of 9th Annual European Symposium on Algorithms (ESA’2001), to appear. 2001.Google Scholar
  6. 6.
    K. Mehlhorn and M. Ziegelmann. Resource constrained shortest paths. In Proceedings of 8th Annual European Symposium on Algorithms (ESA’ 2000), volume 1879 of Lecture Notes in Computer Science, pages 326–337. Springer, 2000.Google Scholar
  7. 7.
    K. Mehlhorn and M. Ziegelmann. CNOP — a package for constrained network optimization. In 3rd Workshop on Algorithm Engineering and Experiments (ALENEX’01). 2001.Google Scholar
  8. 8.
    J. Mote, I. Murthy, and D. L. Olson. A parametric approach to solving bicriterion shortest path problems. European Journal of Operations Research, 53:81–92, 1991.zbMATHCrossRefGoogle Scholar
  9. 9.
    F. Schulz, D. Wagner, and K. Weihe. Dijkstra’s algorithm on-line: An empirical case study from public railroad transport. In Proceedings of 3rd Workshop on Algorithm Engineering (WAE’99), volume 1668 of Lecture Notes in Computer Science, pages 110–123. Springer, 1999.Google Scholar
  10. 10.
    A. J. V. Skriver and K. A. Andersen. A label correcting approach for solving bicriterion shortest path problems. Computers and Operations Research, 27:507–524, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D. Theune. Robuste und effiziente Methoden zur Lösung von Wegproblemen. Teubner Verlag, Stuttgart, 1995.Google Scholar
  12. 12.
    A. Warburton. Approximation of pareto optima in multiple-objective shortest path problems. Operations Research, 35:70–79, 1987.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Matthias Müller-Hannemann
    • 1
  • Karsten Weihe
    • 1
  1. 1.Forschungsinstitut für Diskrete MathematikRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

Personalised recommendations