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

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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.

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  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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. Bischl, B., Mersmann, O., Trautmann, H., Weihs, C.: Resampling methods in model validation. Evol. Comput. J. 20(2), 249–275 (2012)

    Article  Google Scholar 

  5. Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001)

    Article  MATH  Google Scholar 

  6. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont, CA (1984)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Croes, G.A.: A method for solving traveling-salesman problems. Oper. Res. 6(6), 791–812 (1958)

    Article  MathSciNet  Google Scholar 

  9. Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press (2004)

  10. Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Eiben, A., Smith, J.: Introduction to Evolutionary Computing. Springer (2007)

  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)

  13. Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991)

    Article  MATH  Google Scholar 

  14. Glover, F.: Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Appl. Math. 65(1–3), 223–253 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hoos, H.H., Stützle, T.: Stochastic Local Search: Foundations & Applications. Elsevier/Morgan Kaufmann (2004)

  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)

  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. 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. Kohavi, R., John, G.H.: Wrappers for feature subset selection. Artif. Intell. 97(1–2), 273–324 (1997)

    Article  MATH  Google Scholar 

  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 Intelligence

  21. Kovárik, O., Málek, R.: Meta-learning and meta-optimization. Tech. rep., CTU Technical Report KJB2012010501 003, Prague (2012).

  22. van Laarhoven, P., Aarts, E.: Simulated Annealing: Theory and Applications. Springer (1997)

  23. Lin, S.: Computer solutions of the travelling salesman problem. Bell Syst. Tech. J. 44(10), 2245–2269 (1965)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Chapter  Google Scholar 

  27. Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Syst. 13(4), 297–345 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. Algorithmica 54(2), 243–255 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization – Algorithms and Their Computational Complexity. Springer (2010)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  31. R Development Core Team: R: R Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2011). ISBN 3-900051-07-0

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  35. Smith-Miles, K., Lopes, L.: Measuring instance difficulty for combinatorial optimization problems. Comput. OR 39(5), 875–889 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Stadler, P.F., Schnabl, W.: The landscape of the traveling salesman problem. Phys. Lett. A161, 337–344 (1992)

    Article  MathSciNet  Google Scholar 

  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)

  38. Vazirani, V.V.: Approximation Algorithms. Springer (2001)

  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)

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Correspondence to Heike Trautmann.

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The conference version of this article appeared in the proceedings of the Learning and Intelligent Optimization Conference (LION) 2012 [25].

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Mersmann, O., Bischl, B., Trautmann, H. et al. A novel feature-based approach to characterize algorithm performance for the traveling salesperson problem. Ann Math Artif Intell 69, 151–182 (2013).

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