Advertisement

Computational Optimization and Applications

, Volume 74, Issue 3, pp 851–893 | Cite as

An auction-based approach for the re-optimization shortest path tree problem

  • P. Festa
  • F. GuerrieroEmail author
  • A. Napoletano
Article
  • 57 Downloads

Abstract

The shortest path tree problem is one of the most studied problems in network optimization. Given a directed weighted graph, the aim is to find a shortest path from a given origin node to any other node of the graph. When any change occurs (i.e., the origin node is changed, some nodes/arcs are added/removed to/from the graph, the cost of a subset of arcs is increased/decreased), in order to determine a (still) optimal solution, two different strategies can be followed: a re-optimization algorithm is applied starting from the current optimal solution or a new optimal solution is built from scratch. Generally speaking, the Re-optimization Shortest Path Tree Problem (R-SPTP) consists in solving a sequence of shortest path problems, where the kth problem differs only slightly from the \((k-1){th}\) one, by exploiting the useful information available after each shortest path tree computation. In this paper, we propose an exact algorithm for the R-SPTP, in the case of origin node change. The proposed strategy is based on a dual approach, which adopts a strongly polynomial auction algorithm to extend the solution under construction. The approach is evaluated on a large set of test problems. The computational results underline that it is very promising and outperforms or at least is not worse than the solution approaches reported in the literature.

Keywords

Networks Re-optimization Shortest path Auction approach 

Notes

References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Chapter iv network flows. Handb. Oper. Res. Manag. Sci. 1, 211–369 (1989)Google Scholar
  2. 2.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs, NJ (1993) zbMATHGoogle Scholar
  3. 3.
    Bazaraa, M., Langley, R.: A dual shortest path algorithm. SIAM J. Appl. Math. 26(3), 496–501 (1974)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertsekas, D.P.: A Distributed Algorithm for the Assignment Problem. Laboratory for Information and Decision Systems Working Paper, M.I.T., Cambridge, MA (1979)Google Scholar
  5. 5.
    Bertsekas, D.P.: An auction algorithm for shortest paths. SIAM J. Optim. 1(4), 425–447 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bertsekas, D.: Linear Networks Optimization: Algorithms and Codes. MIT Press (1991)Google Scholar
  7. 7.
    Bertsekas, D.P., Tseng, P., et al.: RELAX-IV: A Faster Version of the RELAX Code for Solving Minimum Cost Flow Problems. Massachusetts Institute of Technology, Laboratory for Information and Decision Systems Cambridge, Cambridge (1994)Google Scholar
  8. 8.
    Bertsekas, D.P., Pallottino, S., Scutellà, M.G.: Polynomial auction algorithms for shortest paths. Comput. Optim. Appl. 4(2), 99–125 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bertsekas, D.P., Guerriero, F., Musmanno, R.: Parallel asynchronous label-correcting methods for shortest paths. J. Optim. Theory Appl. 88(2), 297–320 (1996)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Buriol, L.S., Resende, M.G., Thorup, M.: Speeding up dynamic shortest-path algorithms. INFORMS J. Comput. 20(2), 191–204 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cerulli, R., De Leone, R., Piacente, G.: A modified auction algorithm for the shortest path problem. Optim. Methods Softw. 4(3), 209–224 (1994)Google Scholar
  12. 12.
    Cerulli, R., Festa, P., Raiconi, G., Visciano, G.: The auction technique for the sensor based navigation planning of an autonomous mobile robot. J. Intell. Robot. Syst. Theory Appl. 21(4), 373–395 (1998)zbMATHGoogle Scholar
  13. 13.
    Cerulli, R., Festa, P., Raiconi, G.: Shortest paths in randomly time varying networks. In: IEEE Conference on Intelligent Transportation Systems, pp. 854–859 (2001)Google Scholar
  14. 14.
    Cerulli, R., Festa, P., Raiconi, G.: Shortest path auction algorithm without contractions using virtual source concept. Comput. Optim. Appl. 26(2), 191–208 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chan, E.P., Yang, Y.: Shortest path tree computation in dynamic graphs. IEEE Trans. Comput. 58(4), 541–557 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Cherkassky, B.V., Goldberg, A.V., Radzik, T.: Shortest paths algorithms: theory and experimental evaluation. Math. Program. 73(2), 129–174 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28(4), 1326–1346 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Christofides, N.: An Algorithmic Approach. Academic Press Inc., New York (1975)zbMATHGoogle Scholar
  19. 19.
    D’Andrea, A., D’Emidio, M., Frigioni, D., Leucci, S., Proietti, G.: Dynamically maintaining shortest path trees under batches of updates. In: Moscibroda, T., Rescigno, A.A. (eds.) International Colloquium on Structural Information and Communication Complexity, pp. 286–297. Springer (2013) Google Scholar
  20. 20.
    Demetrescu, C., Goldberg, A., Johnson, D.: 9th Dimacs Implementation Challenge-Shortest Paths. American Mathematical Society, Providence (2006)Google Scholar
  21. 21.
    Demetrescu, C., Goldberg, A.V., Johnson, D.S.: The Shortest Path Problem: Ninth DIMACS Implementation Challenge, vol. 74. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  22. 22.
    Di Puglia Pugliese, L., Guerriero, F.: A computational study of solution approaches for the resource constrained elementary shortest path problem. Ann. Oper. Res. 201(1), 131–157 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Di Puglia Pugliese, L., Guerriero, F.: A survey of resource constrained shortest path problems: exact solution approaches. Networks 62(3), 183–200 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dial, R.B.: A probabilistic multipath traffic assignment model which obviates path enumeration. Transp. Res. 5(2), 83–111 (1971)Google Scholar
  25. 25.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ferone, D., Festa, P., Guerriero, F., Laganá, D.: The constrained shortest path tour problem. Comput. Oper. Res. 74, 64–77 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ferone, D., Festa, P., Napoletano, A., Pastore, T.: Shortest paths on dynamic graphs: a survey. Pesqui. Oper. 37(3), 487–508 (2017)Google Scholar
  28. 28.
    Festa, P., Pallottino, S.: A Pseudo-Random Networks Generator. Technical report, Department of Mathematics and Applications “R. Caccioppoli”, University of Napoli FEDERICO II, Italy (2003)Google Scholar
  29. 29.
    Florian, M., Nguyen, S., Pallottino, S.: A dual simplex algorithm for finding all shortest paths. Networks 11(4), 367–378 (1981)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gallo, G.: Reoptimization procedures in shortest path problem. Riv. Mat. Sci. Econ. Soc. 3(1), 3–13 (1980)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gallo, G., Pallottino, S.: A new algorithm to find the shortest paths between all pairs of nodes. Discrete Appl. Math. 4(1), 23–35 (1982).  https://doi.org/10.1016/0166-218X(82)90031-2. ISSN: 0166218XMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Granat, J., Guerriero, F.: The interactive analysis of the multicriteria shortest path problem by the reference point method. Eur. J. Oper. Res. 151(1), 103–118 (2003)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ioachim, I., Gelinas, S., Soumis, F., Desrosiers, J.: A dynamic programming algorithm for the shortest path problem with time windows and linear node costs. Networks 31(3), 193–204 (1998)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jayakrishnan, R., Mahmassani, H.S., Hu, T.-Y.: An evaluation tool for advanced traffic information and management systems in urban networks. Transp. Res. Part C Emerg. Technol. 2(3), 129–147 (1994)Google Scholar
  35. 35.
    Koenig, S., Likhachev, M.: D* lite. In Proceedings of the Eighteenth National Conference on Artificial Intelligence, Edmonton, Alberta, Canada, 28 July 2002–01 August 2002, pp. 476–483.Google Scholar
  36. 36.
    Koenig, S., Likhachev, M.: Fast replanning for navigation in unknown terrain. IEEE Trans. Robot. 21(3), 354–363 (2005)Google Scholar
  37. 37.
    Koenig, S., Likhachev, M., Furcy, D.: Lifelong planning a. Artif. Intell. 155(1–2), 93–146 (2004)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Magnanti, T.L., Wong, R.T.: Network design and transportation planning: models and algorithms. Transp. Sci. 18(1), 1–55 (1984)Google Scholar
  39. 39.
    Nannicini, G., Baptiste, P., Krob, D., Liberti, L.: Fast paths in dynamic road networks. Proc. ROADEF 8, 1–14 (2008)zbMATHGoogle Scholar
  40. 40.
    Nguyen, S., Pallottino, S., Scutellà, M.G.: A new dual algorithm for shortest path reoptimization. In: Gendreau, M., Marcotte, P. (eds.) Transportation and Network Analysis: Current Trends, pp. 221–235. Springer, Boston, MA (2002)Google Scholar
  41. 41.
    Pallottino, S., Scutella, M.G.: Strongly polynomial auction algorithms for shortest paths. Ric. Op. 60, 33–53 (1991)Google Scholar
  42. 42.
    Pallottino, S., Scutellà, M.G.: Dual algorithms for the shortest path tree problem. Networks 26(2), 125–133 (1997)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Pallottino, S., Scutellá, M.G.: Shortest path algorithms in transportation models: classical and innovative aspects. In: Marcotte, P., Nguyen, S. (eds.) Equilibrium and Advanced Transportation Modelling, pp. 245–281. Springer, Boston, MA (1998)zbMATHGoogle Scholar
  44. 44.
    Pallottino, S., Scutellá, M.G.: A new algorithm for reoptimizing shortest paths when the arc costs change. Oper. Res. Lett. 31(2), 149–160 (2003)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Pettie, S., Ramachandran, V.: Command Line Tools Generating Various Families of Random Graphs. American Mathematical Society, Providence (2006)Google Scholar
  46. 46.
    Pham, P.P., Perreau, S.: Performance analysis of reactive shortest path and multipath routing mechanism with load balance. In: INFOCOM 2003. Twenty-Second Annual Joint Conference of the IEEE Computer and Communications. IEEE Societies, vol. 1, pp. 251–259. IEEE (2003)Google Scholar
  47. 47.
    Ramalingam, G., Reps, T.: An incremental algorithm for a generalization of the shortest-path problem. J. Algorithms 21(2), 267–305 (1996)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Schwartz, M.: Telecommunication Networks: Protocols, Modeling and Analysis, vol. 7. Addison-Wesley, Reading (1987)Google Scholar
  49. 49.
    Shi, N., Zhou, S., Wang, F., Tao, Y., Liu, L.: The multi-criteria constrained shortest path problem. Transp. Res. Part E Logist. Transp. Rev. 101, 13–29 (2017)Google Scholar
  50. 50.
    Stentz, A.: Optimal and efficient path planning for partially-known environments. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA ’94), vol. 4, pp. 3310 – 3317 (1994)Google Scholar
  51. 51.
    Stentz, A.: The focussed \(\text{d}^{*}\) algorithm for real-time replanning. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence—Volume 2, IJCAI’95, pp. 1652–1659, San Francisco, CA, USA, 1995. Morgan Kaufmann Publishers Inc. ISBN: 1-55860-363-8. http://dl.acm.org/citation.cfm?id=1643031.1643113
  52. 52.
    Toth, P., Vigo, D.: Vehicle Routing: Problems, Methods, and Applications. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  53. 53.
    Zhang, X., Zhang, Z., Zhang, Y., Wei, D., Deng, Y.: Route selection for emergency logistics management: a bio-inspired algorithm. Saf. Sci. 54, 87–91 (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsUniversity of Napoli Federico IINaplesItaly
  2. 2.Department of Mechanical, Energy and Management EngineeringUniversity of CalabriaRende, CosenzaItaly

Personalised recommendations