, Volume 11, Issue 4, pp 323–348 | Cite as

Speeding up Martins’ algorithm for multiple objective shortest path problems

  • Sofie DemeyerEmail author
  • Jan Goedgebeur
  • Pieter Audenaert
  • Mario Pickavet
  • Piet Demeester
Research paper


The latest transportation systems require the best routes in a large network with respect to multiple objectives simultaneously to be calculated in a very short time. The label setting algorithm of Martins efficiently finds this set of Pareto optimal paths, but sometimes tends to be slow, especially for large networks such as transportation networks. In this article we investigate a number of speedup measures, resulting in new algorithms. It is shown that the calculation time to find the Pareto optimal set can be reduced considerably. Moreover, it is mathematically proven that these algorithms still produce the Pareto optimal set of paths.


Multiobjective shortest path problem Labeling algorithm  Stop condition Bidirectional routing Pareto optimal set 

Mathematics Subject Classification (2000)

90C29 05C85 05C38 68Q25 90B06 


  1. 9th DIMACS implementation challenge (2006) Shortest paths.
  2. Azevedo JA, Costa MEOS, Madeira JJERS, Martins EQV (1993) An algorithm for the ranking of shortest paths. Eur J Oper Res 69:97–106CrossRefGoogle Scholar
  3. Bellman R (1958) On a routing problem. Q Appl Math 16:87–90Google Scholar
  4. Bornstein C, Maculan N, Pascoal MMB, Pinto L (2012) Multiobjective combinatorial optimization problems with a cost and several bottleneck objective functions: an algorithm with reoptimization. Comput Oper Res 39:1969–1976CrossRefGoogle Scholar
  5. Brumbaugh-Smith J, Shier D (1989) An empirical investigation of some bicriterion shortest path algorithms. Eur J Oper Res 43:216–224CrossRefGoogle Scholar
  6. Clímaco JNC, Martins EQV (1982) A bicriterion shortest path algorithm. Eur J Oper Res 11:399–404CrossRefGoogle Scholar
  7. Clímaco JNC, Pascoal MMB (2012) Multicriteria path and tree problems—discussion on exact algorithms and applications. Int Trans Oper Res 19:63–98CrossRefGoogle Scholar
  8. de Lima Pinto L, Bornstein CT, Maculan N (2009) The tricriterion shortest path problem with at least two bottleneck objective functions. Eur J Oper Res 198:387–391CrossRefGoogle Scholar
  9. Demeyer S, Audenaert P, Slock B, Pickavet M, Demeester P (2008) Multimodal transport planning in a dynamic environment. Conference on intelligent public transport systems, Amsterdam, pp 155–167Google Scholar
  10. Dijkstra EW (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1:269–271CrossRefGoogle Scholar
  11. Disser Y, Müller-Hannemann M, Schnee M (2008) Multi-criteria shortest paths in time-dependent train networks, experimental algorithms, 7th international workshop, WEA 2008. Provincetown, MA, USA pp 347–361Google Scholar
  12. Ehrgott M, Gandibleux X (2000) A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum 22:425–460CrossRefGoogle Scholar
  13. Ehrgott M, Gandibleux X (2002) Multiple criteria optimization: state of the art annotated bibliographic surveys, international series in operations research and management science, vol. 52, SpringerGoogle Scholar
  14. Eppstein D (1998) Finding the k shortest paths. SIAM J Comput 28:652–673CrossRefGoogle Scholar
  15. Fu L, Sun D, Rilett LR (2006) Heuristic shortest path algorithms for transportation applications: state of the art. Comput Oper Res 33:3324–3343CrossRefGoogle Scholar
  16. Gandibleux X, Beugnies F, Randriamasy S (2006) Martins’ algorithm revisited for multi-objective shortest path problems with MaxMin cost function. 4OR Q J Oper Res 4:47–59CrossRefGoogle Scholar
  17. Garroppo RG, Giordano S, Tavanti L (2010) A survey on multi-constrained optimal path computation: exact and approximate algorithms. Comput Netw 54:3081–3107CrossRefGoogle Scholar
  18. Geisberger R, Sanders P, Schultes D, Delling D (2008) Contraction Hierarchies: faster and simpler hierarchical routing in road networks. In: McGeoch (ed) Workshop on experimental algorithms, LNCS 5038. Springer-Verlag, Berlin/Heidelberg, pp 319–333Google Scholar
  19. Goldberg A, Kaplan H, Werneck R (2006) Reach for A*: efficient point-to-point shortest path algorithms. In: Workshop on algorithm engineering and experiments, Miami, pp 129–143Google Scholar
  20. Guerriero F, Musmanno R (2001) Label correcting methods to solve multicriteria shortest path problems. J Optim Theory Appl 111:589–613CrossRefGoogle Scholar
  21. Hansen P (1980) Bicriterion path problems. In: Fandel G, Gal T (eds) Multiple criteria decision making: theory and applications, lecture notes in economics and in mathematical systems 177. Springer, Heidelberg, pp 109–127CrossRefGoogle Scholar
  22. Jiménez V, Marzal A (1999) Computing the K shortest paths: a new algorithm and experimental comparison. In: Vitter JS, Zaroliagis CD (eds) Proceedings of the 3rd international workshop on algorithm engineering, LNCS 1668. Springer-Verlag Berlin/Heidelberg, pp 15–29Google Scholar
  23. Jiménez V, Marzal A (2003) A lazy version of Eppstein’s k shortest path algorithm. In: Jansen K, Margraf M, Matrolli M, Rolim J (eds) Proceedings of the 2nd international workshop on experimental and efficient algorithms, LNCS 2647. Springer-Verlag Berlin/Heidelberg, pp 179–191Google Scholar
  24. Martins EQV (1984a) On a multicriteria shortest path problem. Eur J Oper Res 16:236–245CrossRefGoogle Scholar
  25. Martins EQV (1984b) An algorithm for ranking paths that may contain cycles. Eur J Oper Res 18:123–130CrossRefGoogle Scholar
  26. Martins EQV, Paixão JM, Rosa MS, Santos JLE (2007) Ranking multiobjective shortest paths, Pré-publicações do Departamento de Matemática 07–11, Universidade de CoimbraGoogle Scholar
  27. Martins EQV, Pascoal MMB, Santos JLE (1999) Deviation algorithms for ranking shortest paths. Int J Found Comput Sci 10:247–263CrossRefGoogle Scholar
  28. Martins EQV, Pascoal MMB, Santos JLE (2000) Labeling algorithms for ranking shortest paths. Technical Report 001 CISUCGoogle Scholar
  29. Martins EQV, Pascoal MMB, Santos JLE (2001) A new improvement for a K shortest path algorithm. Investigação Oper 21:47–60Google Scholar
  30. Martins EQV, Santos JLE (1999) The labeling algorithm for the multiobjective shortest path problem, CISUC technical report TR 99/005, University of Coimbra, PortugalGoogle Scholar
  31. Nicholson JAT (1966) Finding the shortest route between two points in a network. Comput J 9:275–280CrossRefGoogle Scholar
  32. Pangilinan JMA, Janssens GK (2007) Evolutionary algorithms for the multiobjective shortest path problem. Int J Appl Sci Eng Technol 4:205–210Google Scholar
  33. Paixão JM, Santos JL (2007) Labelling methods for the general case of the multi-objective shortest path problem—a computational study, Pré-publicações do Departamento de Matemática 07-42, Universidade de CoimbraGoogle Scholar
  34. Paixão JM, Santos JL (2008) A new ranking path algorithm for the multiobjective shortest path problem, Pré-publicações do Departamento de Matemática 08-27, Universidade de CoimbraGoogle Scholar
  35. Pinto L, Pascoal MMB (2010) On algorithms for tricriteria shortest path problems with two bottleneck objective functions. Comput Oper Res 37:1774–1779CrossRefGoogle Scholar
  36. Raith A (2010) Speed-up of labelling algorithms for biobjective shortest path problems. In: Proceedings of the 45th annual conference of the ORSNZ. Auckland, New Zealand, pp 313–322Google Scholar
  37. Raith A, Ehrgott M (2009) A comparison of solution strategies for biobjective shortest path problems. Comput Oper Res 36:1299–1331CrossRefGoogle Scholar
  38. Sastry V, Janakiraman T, Mohideen S (2003) New algorithms for multi-objective shortest path problem. Opsearch 40:278–298Google Scholar
  39. Serafini P (1986) Some considerations about computational complexity for multiobjective combinatorial problems. Recent advances and historical development of vector optimization 294:222–232CrossRefGoogle Scholar
  40. Skriver AJV (2000) A classification of bicriterion shortest path (BSP) algorithms. Asia Pac J Oper Res 17:192–212Google Scholar
  41. Skriver AJV, Andersen K (2000) A label correcting approach for solving bicriterion shortest-path problems. Comput Oper Res 27:507–524CrossRefGoogle Scholar
  42. Stewart BS, White CC (1991) Multiobjective A*. J Assoc Comput Mach 38:775–814CrossRefGoogle Scholar
  43. Vincke P (1974) Problemes multicritères. Cahiers Centre Etudes Recherche Operationnelle 16:425–439Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sofie Demeyer
    • 1
    Email author
  • Jan Goedgebeur
    • 1
  • Pieter Audenaert
    • 1
  • Mario Pickavet
    • 1
  • Piet Demeester
    • 1
  1. 1.Department of Information Technology (INTEC)Ghent University, IBBTGhentBelgium

Personalised recommendations