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On the cardinality of the Pareto set in bicriteria shortest path problems

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  • Matthias Müller-Hannemann
  • Karsten Weihe

Abstract

Computing shortest paths with two or more conflicting optimization criteria is a fundamental problem in transportation and logistics. 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. For typical characteristics which occur in various applications we study in this paper whether we can bound the size of the Pareto set to a polynomial size or not. These characteristics are also evaluated (1) on a concrete application scenario (computing the set of best train connections in view of travel time, fare, and number of train changes) and (2) on a simplified randomized model. It will turn out that the number of Pareto optima on each visited node is restricted by a small constant in our concrete application, and that the size of the Pareto set is much smaller than our worst case bounds in the randomized model. Copyright Springer Science + Business Media, LLC 2006

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  • Matthias Müller-Hannemann & Karsten Weihe, 2006. "On the cardinality of the Pareto set in bicriteria shortest path problems," Annals of Operations Research, Springer, vol. 147(1), pages 269-286, October.
    • Handle: RePEc:spr:annopr:v:147:y:2006:i:1:p:269-286:10.1007/s10479-006-0072-1
    DOI: 10.1007/s10479-006-0072-1
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    1. Mote, John & Murthy, Ishwar & Olson, David L., 1991. "A parametric approach to solving bicriterion shortest path problems," European Journal of Operational Research, Elsevier, vol. 53(1), pages 81-92, July.
    2. Arthur Warburton, 1987. "Approximation of Pareto Optima in Multiple-Objective, Shortest-Path Problems," Operations Research, INFORMS, vol. 35(1), pages 70-79, February.
    3. Brumbaugh-Smith, J. & Shier, D., 1989. "An empirical investigation of some bicriterion shortest path algorithms," European Journal of Operational Research, Elsevier, vol. 43(2), pages 216-224, November.
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    1. Pascal Halffmann & Tobias Dietz & Anthony Przybylski & Stefan Ruzika, 2020. "An inner approximation method to compute the weight set decomposition of a triobjective mixed-integer problem," Journal of Global Optimization, Springer, vol. 77(4), pages 715-742, August.
    2. Andrew Ensor & Felipe Lillo, 2016. "Colored-Edge Graph Approach for the Modeling of Multimodal Transportation Systems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(01), pages 1-21, February.
    3. Pulido, Francisco Javier & Mandow, Lawrence & Pérez de la Cruz, José Luis, 2014. "Multiobjective shortest path problems with lexicographic goal-based preferences," European Journal of Operational Research, Elsevier, vol. 239(1), pages 89-101.
    4. Vidal, Thibaut & Laporte, Gilbert & Matl, Piotr, 2020. "A concise guide to existing and emerging vehicle routing problem variants," European Journal of Operational Research, Elsevier, vol. 286(2), pages 401-416.
    5. Enrique Machuca & Lawrence Mandow, 2016. "Lower bound sets for biobjective shortest path problems," Journal of Global Optimization, Springer, vol. 64(1), pages 63-77, January.
    6. Ehrgott, Matthias & Wang, Judith Y.T. & Raith, Andrea & van Houtte, Chris, 2012. "A bi-objective cyclist route choice model," Transportation Research Part A: Policy and Practice, Elsevier, vol. 46(4), pages 652-663.
    7. Duque, Daniel & Lozano, Leonardo & Medaglia, Andrés L., 2015. "An exact method for the biobjective shortest path problem for large-scale road networks," European Journal of Operational Research, Elsevier, vol. 242(3), pages 788-797.
    8. Ehsan Jafari & Stephen D. Boyles, 2017. "Multicriteria Stochastic Shortest Path Problem for Electric Vehicles," Networks and Spatial Economics, Springer, vol. 17(3), pages 1043-1070, September.
    9. Diclehan Tezcaner Öztürk & Murat Köksalan, 2016. "An interactive approach for biobjective integer programs under quasiconvex preference functions," Annals of Operations Research, Springer, vol. 244(2), pages 677-696, September.
    10. Breugem, T. & Dollevoet, T.A.B. & van den Heuvel, W., 2016. "Analysis of FPTASes for the Multi-Objective Shortest Path Problem," Econometric Institute Research Papers EI2016-03, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    11. Kuhn, K. & Raith, A. & Schmidt, M. & Schöbel, A., 2016. "Bi-objective robust optimisation," European Journal of Operational Research, Elsevier, vol. 252(2), pages 418-431.
    12. Machuca, E. & Mandow, L. & Pérez de la Cruz, J.L. & Ruiz-Sepulveda, A., 2012. "A comparison of heuristic best-first algorithms for bicriterion shortest path problems," European Journal of Operational Research, Elsevier, vol. 217(1), pages 44-53.

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