Advertisement

Journal of Heuristics

, Volume 20, Issue 1, pp 107–124 | Cite as

A backbone based TSP heuristic for large instances

  • Gerold Jäger
  • Changxing Dong
  • Boris Goldengorin
  • Paul Molitor
  • Dirk Richter
Article

Abstract

We introduce a reduction technique for large instances of the traveling salesman problem (TSP). This approach is based on the observation that tours with good quality are likely to share many edges. We exploit this observation by neglecting the less important tour space defined by the shared edges, and searching the important tour subspace in more depth. More precisely, by using a basic TSP heuristic, we obtain a set of starting tours. We call the set of edges which are contained in each of these starting tours as pseudo-backbone edges. Then we compute the maximal paths consisting only of pseudo-backbone edges, and transform the TSP instance to another one with smaller size by contracting each such path to a single edge. This reduced TSP instance can be investigated more intensively, and each tour of the reduced instance can be expanded to a tour of the original instance. Combining our reduction technique with the currently leading TSP heuristic of Helsgaun, we experimentally investigate 32 difficult VLSI instances from the well-known TSP homepage. In our experimental results we set world records for seven VLSI instances, i.e., find better tours than the best tours known so far (two of these world records have since been improved upon by Keld Helsgaun and Yuichi Nagata, respectively). For the remaining instances we find tours that are equally good or only slightly worse than the world record tours.

Keywords

Traveling salesman problem Lin-Kernighan Heuristic  Helsgaun Heuristic (LKH) Pseudo-Backbones 

Notes

Acknowledgments

This work was supported by German Research Foundation (DFG) with the Grant Number MO 645/7-3. Boris Goldengorin’s research was partially supported by the Exchange Visitor Program Number: P-1-01285 at Center for Applied Optimization (CAO), University of Florida.

Supplementary material

10732_2013_9233_MOESM1_ESM.gz (1430.1 mb)
Supplementary material 1 (gz 1464379 KB)

References

  1. Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)Google Scholar
  2. Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J., Espinoza, D., Goycoolea, M., Helsgaun, K.: Certification of an optimal tour through 85900 cities. Oper. Res. Lett. 37(1), 11–15 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. Applegate, D., Cook, W., Rohe, A.: Chained Lin-Kernighan for large traveling salesman problems. INFORMS J. Comput. 15(1), 82–92 (2003)CrossRefMATHMathSciNetGoogle Scholar
  4. Balas, E., Simonetti, N.: Linear time dynamic programming algorithms for new classes of restricted TSPs: a computational study. INFORMS J. Comput. 13, 56–75 (2001)CrossRefMATHMathSciNetGoogle Scholar
  5. Bekker, H., Braad, E.P., Goldengorin, B., et al.: Using bipartite and multidimensional matching to select the roots of a system of polynomial equation. In: Gervasi, O. (ed.) Computational Science and Its Applications-ICCSA, Lecture Notes in Computer Science, pp. 397–406. Springer, Berlin (2005)Google Scholar
  6. Climer, S., Zhang, W.: Searching for Backbones and Fat: A Limit-Crossing Approach with Applications. Proceedings of the 18th National Conference on Artificial Intelligence (AAAI), pp. 707–712. AAAI Press, Menlo Park (2002).Google Scholar
  7. Cook, W.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press, Princeton (2011)Google Scholar
  8. Cook, W., Seymour, P.: Tour merging via branch-decomposition. INFORMS J. Comput. 15(3), 233–248 (2003)CrossRefMATHMathSciNetGoogle Scholar
  9. Ernst, C., Dong, C., Jäger, G., Molitor, P., Richter, D.: Finding good tours for huge Euclidean TSP instances by iterative backbone contraction. In: Chen, B. (ed.) AAIM 2010, Lecture Notes in Computer Science, pp. 119–130. Springer, Berlin (2010)Google Scholar
  10. Fischer, T., Merz, P.: Reducing the size of traveling salesman problem instances by fixing edges. EvoCOP 2007, Lecture Notes in Computer Science, pp. 72–83. Springer, Berlin (2007)Google Scholar
  11. Gamboa, D., Rego, C., Glover, F.: Data structures and ejection chains for solving large scale traveling salesman problems. Eur. J. Oper. Res. 160(1), 154–171 (2005)CrossRefMATHGoogle Scholar
  12. Gamboa, D., Rego, C., Glover, F.: Implementation analysis of efficient heuristic algorithms for the traveling salesman problem. Comput. Oper. Res. 33(4), 1154–1172 (2006)CrossRefMATHGoogle Scholar
  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of \({\cal NP}\)-Completeness. Series of Books in the Mathematical Sciences. W.H. Freeman and Company, San Francisco (1979)Google Scholar
  14. Germs, R., Goldengorin, B., Turkensteen, M.: Lower tolerance-based branch and bound algorithms for the ATSP. Comput. Oper. Res. 39(2), 291–298 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. Goldengorin, B., Jäger, G., Molitor, P.: Some Basics on tolerances. In: Cheng, S.-W., Poon, C.K. (eds.) AAIM 2006, Lecture Notes in Computer Science, pp. 194–206. Springer, Berlin (2006a)Google Scholar
  16. Goldengorin, B., Jäger, G., Molitor, P.: Tolerances applied in combinatorial optimization. J. Comput. Sci. 2(9), 716–734 (2006b)CrossRefGoogle Scholar
  17. Gutin, G., Punnen, A.P. (eds.): The Traveling Salesman Problem and Its Variations. Kluwer, Dordrecht (2002)MATHGoogle Scholar
  18. Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)CrossRefMATHMathSciNetGoogle Scholar
  19. Helsgaun, K.: General \(k\)-opt submoves for the Lin-Kernighan TSP heuristic. Math. Program. Comput. 1(2–3), 119–163 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. Johnson, D., McGeoch, L.: The traveling salesman problem: a case study in local optimization. In: Aarts, E., Lenstra, J.K. (eds.) Local Search in Combinatorial Optimization, pp. 215–310. Wiley, Chicester (1997)Google Scholar
  21. Kilby, P., Slaney, J.K., Walsh, T.: The backbone of the travelling salesperson. In: Kaelbling, L.P., Saffiotti A. (Eds.): Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI-05), pp. 175–180, 2005.Google Scholar
  22. Lawler, E.L., Lenstra, J.K., Rinnooy-Kan, A.H.G., Shmoys, D.B. (eds.): The traveling salesman problem—a guided tour of combinatorial optimization. Wiley, Chicester (1985)MATHGoogle Scholar
  23. Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)CrossRefMATHMathSciNetGoogle Scholar
  24. Martin, O., Otto, S.W., Felten, E.W.: Large-step Markov chains for the traveling salesman problem. Complex Syst. 5(3), 299–326 (1991)MATHMathSciNetGoogle Scholar
  25. Martin, O., Otto, S.W., Felten, E.W.: Large-step Markov chains for the TSP incorporating local search heuristics. Oper. Res. Lett. 11, 219–224 (1992)CrossRefMATHMathSciNetGoogle Scholar
  26. Möbius, A., Freisleben, B., Merz, P., Schreiber, M.: Combinatorial optimization by iterative partial transcription. Phys. Rev. E 59(4), 4667–4674 (1999)CrossRefGoogle Scholar
  27. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyanski, L.: Determining computational complexity for characteristic phase transitions. Nature 400, 133–137 (1998)Google Scholar
  28. Richter, D.: Toleranzen in Helsgauns Lin-Kernighan-Heuristik für das TSP. Diploma Thesis, University of Halle-Wittenberg, Germany (2006)Google Scholar
  29. Richter, D., Goldengorin, B., Jäger, G., Molitor, P.: Improving the efficiency of Helsgaun’s Lin-Kernighan heuristic for the symmetric TSP. In: Janssen, J., Pralat, P. (eds.) CAAN 2007, Lecture Notes in Computer Science, pp. 99–111. Springer, Berlin (2007)Google Scholar
  30. Schilham, R.M.F.: Commonalities in Local Search. Ph.D. Thesis, Technische Universiteit Eindhoven, The Netherlands (2001)Google Scholar
  31. Slaney, J.K., Walsh, T.: The backbones in optimization and approximation. Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI-01), pp. 254–259. Kaufmann Publishers, San Francisco (2001)Google Scholar
  32. Tamaki, H.: Alternating cycles contribution: a tour merging strategy for the traveling salesman problem. Research Report MPI-I-2003-1-007, Max-Planck-Institut für Informatik, Saarbrücken, Germany (2003)Google Scholar
  33. Turkensteen, M., Ghosh, D., Goldengorin, B., Sierksma, G.: Tolerance-based branch and bound algorithms for the ATSP. Eur. J. Oper. Res. 189(3), 775–788 (2008)CrossRefMATHMathSciNetGoogle Scholar
  34. Walshaw, C.: A multilevel approach to the traveling salesman problem. Oper. Res. 50(5), 862–877 (2002)CrossRefMATHMathSciNetGoogle Scholar
  35. Zhang, W., Looks, M.: A novel local search Algorithm for the traveling salesman problem that exploits backbones. In Kaelbling, L.P., Saffiotti, A. (eds.) Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI-05), pp. 343–350 (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Gerold Jäger
    • 1
  • Changxing Dong
    • 2
  • Boris Goldengorin
    • 3
    • 4
  • Paul Molitor
    • 5
  • Dirk Richter
    • 5
  1. 1.Department of Mathematics and Mathematical StatisticsUniversity of UmeåUmeåSweden
  2. 2.Leibniz Institute of Agricultural Development in Central and Eastern EuropeHalleGermany
  3. 3.Department of Industrial and Systems EngineeringCenter for Applied Optimization, University of FloridaGainesvilleUSA
  4. 4.Department of OperationsUniversity of GroningenGroningenThe Netherlands
  5. 5.Computer Science InstituteUniversity of Halle-WittenbergHalleGermany

Personalised recommendations