Local Search and the Traveling Salesman Problem: A Feature-Based Characterization of Problem Hardness

  • Olaf Mersmann
  • Bernd Bischl
  • Jakob Bossek
  • Heike Trautmann
  • Markus Wagner
  • Frank Neumann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7219)


With this paper we contribute to the understanding of the success of 2-opt based local search algorithms for solving the traveling salesman problem (TSP). Although 2-opt is widely used in practice, it is hard to understand its success from a theoretical perspective. We take a statistical approach and examine the features of TSP instances that make the problem either hard or easy to solve. As a measure of problem difficulty for 2-opt we use the approximation ratio that it achieves on a given instance. Our investigations point out important features that make TSP instances hard or easy to be approximated by 2-opt.


TSP 2-opt Classification Feature Selection MARS 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Applegate, D., Cook, W.J., Dash, S., Rohe, A.: Solution of a min-max vehicle routing problem. Informs Journal on Computing 14(2), 132–143 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont (1984)zbMATHGoogle Scholar
  4. 4.
    Chandra, B., Karloff, H.J., Tovey, C.A.: New results on the old k-Opt algorithm for the traveling salesman problem. SIAM J. Comput. 28(6), 1998–2029 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Croes, G.A.: A method for solving traveling-salesman problems. Operations Research 6(6), 791–812 (1958)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the tsp: extended abstract. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 1295–1304. SIAM (2007)Google Scholar
  7. 7.
    Friedman, J.H.: Multivariate adaptive regression splines. Annals of Statistics 19(1), 1–67 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Glover, F.: Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Applied Mathematics 65(1-3), 223–253 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Johnson, D.S., McGeoch, L.A.: The traveling salesman problem: A case study in local optimization. In: Aarts, E.H.L., Lenstra, J.K. (eds.) Local Search in Combinatorial Optimization. Wiley (1997)Google Scholar
  10. 10.
    Kanda, J., Carvalho, A., Hruschka, E., Soares, C.: Selection of algorithms to solve traveling salesman problems using meta-learning. Hybrid Intelligent Systems 8, 117–128 (2011)Google Scholar
  11. 11.
    Kilby, P., Slaney, J., Walsh, T.: The backbone of the travelling salesperson. In: Proc, of the 19th International Joint Conference on Artificial intelligence, IJCAI 2005, pp. 175–180. Morgan Kaufmann Publishers Inc., San Francisco (2005)Google Scholar
  12. 12.
    Kötzing, T., Neumann, F., Röglin, H., Witt, C.: Theoretical Properties of Two ACO Approaches for the Traveling Salesman Problem. In: Dorigo, M., Birattari, M., Di Caro, G.A., Doursat, R., Engelbrecht, A.P., Floreano, D., Gambardella, L.M., Groß, R., Şahin, E., Sayama, H., Stützle, T. (eds.) ANTS 2010. LNCS, vol. 6234, pp. 324–335. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Lin, S., Kernighan, B.: An effective heuristic algorithm for the traveling salesman problem. Operations Research 21, 498–516 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lin, S.: Computer solutions of the travelling salesman problem. Bell Systems Technical Journal 44(10), 2245–2269 (1965)zbMATHCrossRefGoogle Scholar
  15. 15.
    Mersmann, O., Bischl, B., Trautmann, H., Preuss, M., Weihs, C., Rudolph, G.: Exploratory landscape analysis. In: Proc. of the 13th Annual Conference on Genetic and Evolutionary Computation, GECCO 2011, pp. 829–836. ACM, New York (2011)Google Scholar
  16. 16.
    Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Systems 13(4), 297–345 (2001)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAMR 33(1), 60–100 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Reinelt, G.: Tsplib - a traveling salesman problem library. ORSA Journal on Computing 3(4), 376–384 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sander, J., Ester, M., Kriegel, H., Xu, X.: Density-based clustering in spatial databases: The algorithm gdbscan and its applications. Data Mining and Knowledge Discovery 2(2), 169–194 (1998)CrossRefGoogle Scholar
  20. 20.
    Smith-Miles, K., van Hemert, J.: Discovering the suitability of optimisation algorithms by learning from evolved instances. Annals of Mathematics and Artificial Intelligence (2011) (forthcoming)Google Scholar
  21. 21.
    Smith-Miles, K., van Hemert, J., Lim, X.Y.: Understanding TSP Difficulty by Learning from Evolved Instances. In: Blum, C., Battiti, R. (eds.) LION 4. LNCS, vol. 6073, pp. 266–280. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  22. 22.
    Stadler, P.F., Schnabl, W.: The Landscape of the Traveling Salesman Problem. Physics Letters A 161, 337–344 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Vazirani, V.V.: Approximation algorithms. Springer (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Olaf Mersmann
    • 1
  • Bernd Bischl
    • 1
  • Jakob Bossek
    • 1
  • Heike Trautmann
    • 1
  • Markus Wagner
    • 2
  • Frank Neumann
    • 2
  1. 1.Statistics FacultyTU Dortmund UniversityGermany
  2. 2.School of Computer ScienceThe University of AdelaideAustralia

Personalised recommendations