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Solving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun Algorithm

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Abstract

The equality generalized traveling salesman problem (E-GTSP) is an extension of the traveling salesman problem (TSP) where the set of cities is partitioned into clusters, and the salesman has to visit every cluster exactly once. It is well known that any instance of E-GTSP can be transformed into a standard asymmetric instance of the TSP, and therefore solved with a TSP solver. This paper evaluates the performance of the state-of-the art TSP solver Lin–Kernighan–Helsgaun (LKH) on transformed E-GTSP instances. Although LKH is used without any modifications, the computational evaluation shows that all instances in a well-known library of benchmark instances, GTSPLIB, could be solved to optimality in a reasonable time. In addition, it was possible to solve a series of new very-large-scale instances with up to 17,180 clusters and 85,900 vertices. Optima for these instances are not known but it is conjectured that LKH has been able to find solutions of a very high quality. The program’s performance has also been evaluated on a large number of instances generated by transforming arc routing problem instances into E-GTSP instances. The program is free of charge for academic and non-commercial use and can be downloaded in source code.

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References

  1. Laporte, G., Asef-Vaziri, A., Sriskandarajah, C.: Some applications of the generalized travelling salesman problem. J. Oper. Res. Soc. 47(12), 1461–1467 (1996)

    Article  MATH  Google Scholar 

  2. Noon, C.E., Bean, J.C.: An efficient transformation of the generalized traveling salesman problem. INFOR 31(1), 39–44 (1993)

    MATH  Google Scholar 

  3. 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)

  4. Ben-Arieh, D., Gutin, G., Penn, M., Yeo, A., Zverovitch, A.: Transformations of generalized ATSP into ATSP. Oper. Res. Lett. 31(5), 357–365 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Laporte, G., Semet, F.: Computational evaluation of a transformation procedure for the symmetric generalized traveling salesman problem. INFOR 37(2), 114–120 (1999)

    Google Scholar 

  6. Karapetyan, D., Gutin, G.: Efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem. Eur. J. Oper. Res. 219(2), 234–251 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Helsgaun, K.: An effective implementation of the Lin–Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Helsgaun, K.: General \(k\)-opt submoves for the Lin–Kernighan TSP heuristic. Math. Prog. Comput. 1(2–3), 119–163 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling salesman problem. Oper. Res. 21(2), 498–516 (1973)

    Article  MathSciNet  Google Scholar 

  10. Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991)

    Article  MATH  Google Scholar 

  11. Fischetti, M., Salazar González, J.J., Toth, P.: A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Oper. Res. 45(3), 378–394 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fischetti, M., SalazarGonzález, J.J., Toth, P.: The generalized traveling salesman and orientering problems. In: Gutin, G., Punnen, A.P. (eds.) The Traveling Salesman Problem and its Variations, pp. 602–662. Kluwer, Dordrecht (2002)

  13. Johnson, D.S., McGeoch, L.A., Glover, F., Rego, C.: 8th DIMACS implementation challenge: the traveling salesman problem (2000). http://dimacs.rutgers.edu/Challenges/TSP/. Accessed 24 Apr 2015

  14. National traveling salesman problems. http://www.math.uwaterloo.ca/tsp/world/countries.html. Accessed 24 Apr 2015

  15. Held, M., Karp, R.M.: The traveling salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gutin, G., Karapetyan, D.: A memetic algorithm for the generalized traveling salesman problem. Nat. Comput. 9(1), 47–60 (2010)

    Article  MathSciNet  Google Scholar 

  17. Karapetyan, D., Gutin, G.: Lin–Kernighan heuristic adaptations for the generalized traveling salesman problem. Eur. J. Oper. Res. 208(3), 221–232 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Blais, M., Laporte, G.: Exact solution of the generalized routing problem through graph transformations. J. Oper. Res. Soc. 54(8), 906–910 (2003)

    Article  MATH  Google Scholar 

  19. Corberán, Á., Plana I., Sanchis, J.M.: Arc routing problems: data instances. http://www.uv.es/corberan/instancias.htm. Accessed 24 Apr 2015

  20. Corberán, A., Oswald, M., Plana, I., Reinelt, G., Sanchis, J.M.: New results on the Windy Postman problem. Math. Program. Ser. A 132, 309–332 (2012)

    Article  MATH  Google Scholar 

  21. Drexl, M.: On the generalized directed rural postman problem. J. Oper. Res. Soc. (2013). doi:10.1057/jors.2013.60

  22. Karapetyan, D., Gutin, G.: Generalized traveling salesman problem reduction algorithms. Algorithm Oper. Res. 4, 144–154 (2009)

    MATH  MathSciNet  Google Scholar 

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  1. Computer Science, Roskilde University, 4000, Roskilde, Denmark

    Keld Helsgaun

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  1. Keld Helsgaun
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Correspondence to Keld Helsgaun.

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Helsgaun, K. Solving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun Algorithm. Math. Prog. Comp. 7, 269–287 (2015). https://doi.org/10.1007/s12532-015-0080-8

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  • DOI: https://doi.org/10.1007/s12532-015-0080-8

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