Exact Graph Search Algorithms for Generalized Traveling Salesman Path Problems

  • Michael N. Rice
  • Vassilis J. Tsotras
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7276)

Abstract

The Generalized Traveling Salesman Path Problem (GTSPP) involves finding the shortest path from a location s to a location t that passes through at least one location from each of a set of generalized location categories (e.g., gas stations, grocery stores). This NP-hard problem type has many applications in transportation and location-based services. We present two exact algorithms for solving GTSPP instances, which rely on a unique product-graph search formulation. Our exact algorithms are exponential only in the number of categories (not in the total number of locations) and do not require the explicit construction of a cost matrix between locations, thus allowing us to efficiently solve many real-world problems to optimality. Experimental analysis on the road network of North America demonstrates that we can optimally solve large-scale, practical GTSPP instances typically in a matter of seconds, depending on the overall number and sizes of the categories.

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References

  1. 1.
    Behzad, A., Modarres, M.: A new efficient transformation of generalized traveling salesman problem into traveling salesman problem. In: Proceedings of the 15th International Conference of Systems Engineering, ICSE (2002)Google Scholar
  2. 2.
    Delling, D., Goldberg, A.V., Nowatzyk, A., Werneck, R.F.F.: Phast: Hardware-accelerated shortest path trees. In: IPDPS, pp. 921–931 (2011)Google Scholar
  3. 3.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dimitrijevic, V., Saric, Z.: An efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs. Inf. Sci. 102(1-4), 105–110 (1997)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fischetti, M., Gonzalez, J.J.S., Toth, P.: A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Operations Research 45(3), 378–394 (1997)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Geisberger, R., Sanders, P., Schultes, D., Delling, D.: Contraction Hierarchies: Faster and Simpler Hierarchical Routing in Road Networks. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 319–333. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Henry-Labordere, A.L.: The record balancing problem: A dynamic programming solution of a generalized traveling salesman problem. RIRO B-2, 43–49 (1969)Google Scholar
  8. 8.
    Laporte, G., Asef-Vaziri, A., Sriskandarajah, C.: Some applications of the generalized travelling salesman problem. The Journal of the Operational Research Society 47(12), 1461–1467 (1996)MATHGoogle Scholar
  9. 9.
    Laporte, G., Mercure, H., Nobert, Y.: Generalized travelling salesman problem through n sets of nodes: The asymmetrical case. Discrete Applied Mathematics 18(2), 185–197 (1987)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Laporte, G., Nobert, Y.: Generalized travelling salesman problem through n sets of nodes: An integer programming approach. INFOR 21(1), 61–75 (1983)MATHGoogle Scholar
  11. 11.
    Lien, Y.-N., Ma, Y.W.E., Wah, B.W.: Transformation of the generalized traveling-salesman problem into the standard traveling-salesman problem. Inf. Sci. 74(1-2), 177–189 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Maue, J., Sanders, P., Matijevic, D.: Goal Directed Shortest Path Queries Using Precomputed Cluster Distances. In: Àlvarez, C., Serna, M. (eds.) WEA 2006. LNCS, vol. 4007, pp. 316–327. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Noon, C.E., Bean, J.C.: A lagrangian based approach for the asymmetric generalized traveling salesman problem. Operations Research 39(4), 623–632 (1991)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Saksena, J.P.: Mathematical model of scheduling clients through welfare agencies. CORS Journal 8, 185–200 (1970)MathSciNetGoogle Scholar
  15. 15.
    Srivastava, S.S., Kumar, S., Garg, R.C., Sen, P.: Generalized travelling salesman problem through n sets of nodes. CORS Journal 7, 97–101 (1969)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael N. Rice
    • 1
  • Vassilis J. Tsotras
    • 1
  1. 1.University of California, Riverside (UCR)RiversideUSA

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