Advertisement

A novel feature-based approach to characterize algorithm performance for the traveling salesperson problem

  • Olaf Mersmann
  • Bernd Bischl
  • Heike Trautmann
  • Markus Wagner
  • Jakob Bossek
  • Frank Neumann
Article

Abstract

Meta-heuristics are frequently used to tackle NP-hard combinatorial optimization problems. With this paper we contribute to the understanding of the success of 2-opt based local search algorithms for solving the traveling salesperson 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.

Keywords

TSP 2-opt Classification Feature selection MARS 

Mathematics Subject Classifications (2010)

90B06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Applegate, D., Cook, W.J., Dash, S., Rohe, A.: Solution of a min-max vehicle routing problem. INFORMS J. Comput. 14(2), 132–143 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bischl, B., Mersmann, O., Trautmann, H., Preuss, M.: Algorithm selection based on exploratory landscape analysis and cost-sensitive learning. In: Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation, GECCO ’12, pp. 313–320. ACM, New York, NY, USA (2012)Google Scholar
  4. 4.
    Bischl, B., Mersmann, O., Trautmann, H., Weihs, C.: Resampling methods in model validation. Evol. Comput. J. 20(2), 249–275 (2012)CrossRefGoogle Scholar
  5. 5.
    Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont, CA (1984)Google Scholar
  7. 7.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Croes, G.A.: A method for solving traveling-salesman problems. Oper. Res. 6(6), 791–812 (1958)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press (2004)Google Scholar
  10. 10.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Eiben, A., Smith, J.: Introduction to Evolutionary Computing. Springer (2007)Google Scholar
  12. 12.
    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
  13. 13.
    Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    Glover, F.: Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Appl. Math. 65(1–3), 223–253 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hoos, H.H., Stützle, T.: Stochastic Local Search: Foundations & Applications. Elsevier/Morgan Kaufmann (2004)Google Scholar
  16. 16.
    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
  17. 17.
    Kanda, J., Carvalho, A., Hruschka, E., Soares, C.: Selection of algorithms to solve traveling salesman problems using meta-learning. IJHIS 8(3), 117–128 (2011)Google Scholar
  18. 18.
    Kilby, P., Slaney, J., Walsh, T.: The backbone of the travelling salesperson. In: Proc, of the 19th International Joint Conference on Artificial Intelligence, IJCAI’05, pp. 175–180. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2005)Google Scholar
  19. 19.
    Kohavi, R., John, G.H.: Wrappers for feature subset selection. Artif. Intell. 97(1–2), 273–324 (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kötzing, T., Neumann, F., Röglin, H., Witt, C.: Theoretical properties of two ACO approaches for the traveling salesman problem. In: Proc. of ANTS 2010, LNCS, vol. 6234, pp. 324–335 (2010). Extended journal version appears in Swarm IntelligenceGoogle Scholar
  21. 21.
    Kovárik, O., Málek, R.: Meta-learning and meta-optimization. Tech. rep., CTU Technical Report KJB2012010501 003, Prague (2012). http://cig.felk.cvut.cz/research/publications/Meta-learning_and_meta-optimization.pdf
  22. 22.
    van Laarhoven, P., Aarts, E.: Simulated Annealing: Theory and Applications. Springer (1997)Google Scholar
  23. 23.
    Lin, S.: Computer solutions of the travelling salesman problem. Bell Syst. Tech. J. 44(10), 2245–2269 (1965)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lin, S., Kernighan, B.: An effective heuristic algorithm for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Mersmann, O., Bischl, B., Bossek, J., Trautmann, H., Wagner, M., Neumann, F.: Local search and the traveling salesman problem: A feature-based characterization of problem hardness. In: Hamadi, Y., Schoenauer, M. (eds.) Learning and Intelligent Optimization. Lecture Notes in Computer Science, pp. 115–129. Springer Berlin Heidelberg (2012)Google Scholar
  26. 26.
    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 ’11, pp. 829–836. ACM, New York, NY, USA (2011)CrossRefGoogle Scholar
  27. 27.
    Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Syst. 13(4), 297–345 (2001)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. Algorithmica 54(2), 243–255 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization – Algorithms and Their Computational Complexity. Springer (2010)Google Scholar
  30. 30.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    R Development Core Team: R: R Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2011). http://www.R-project.org. ISBN 3-900051-07-0
  32. 32.
    Sander, J., Ester, M., Kriegel, H., Xu, X.: Density-based clustering in spatial databases: The algorithm gdbscan and its applications. Data Mining Knowl. Discov. 2(2), 169–194 (1998)CrossRefGoogle Scholar
  33. 33.
    Smith-Miles, K., van Hemert, J.: Discovering the suitability of optimisation algorithms by learning from evolved instances. Ann. Math. Artif. Intell. 61(2), 87–104 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Smith-Miles, K., van Hemert, J.I., Lim, X.Y.: Understanding tsp difficulty by learning from evolved instances. In: Blum, C., Battiti, R. (eds.) LION, vol. 6073, pp. 266–280. Lecture Notes in Computer Science. Springer (2010)Google Scholar
  35. 35.
    Smith-Miles, K., Lopes, L.: Measuring instance difficulty for combinatorial optimization problems. Comput. OR 39(5), 875–889 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Stadler, P.F., Schnabl, W.: The landscape of the traveling salesman problem. Phys. Lett. A161, 337–344 (1992)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Sutton, A.M., Neumann, F.: A parameterized runtime analysis of evolutionary algorithms for the euclidean traveling salesperson problem. In: Hoffmann, J., Selman, B. (eds.) AAAI. AAAI Press (2012)Google Scholar
  38. 38.
    Vazirani, V.V.: Approximation Algorithms. Springer (2001)Google Scholar
  39. 39.
    Wegener, I.: Simulated annealing beats Metropolis in combinatorial optimization. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP ’05), vol. 3580, pp. 589–601. Lecture Notes on Computer Science. Springer (2005)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

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

Personalised recommendations