Advertisement

The Multi-Funnel Structure of TSP Fitness Landscapes: A Visual Exploration

  • Gabriela Ochoa
  • Nadarajen Veerapen
  • Darrell Whitley
  • Edmund K. Burke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9554)

Abstract

We use the Local Optima Network model to study the structure of symmetric TSP fitness landscapes. The ‘big-valley’ hypothesis holds that for TSP and other combinatorial problems, local optima are not randomly distributed, instead they tend to be clustered around the global optimum. However, a recent study has observed that, for solutions close in evaluation to the global optimum, this structure breaks down into multiple valleys, forming what has been called ‘multiple funnels’. The multiple funnel concept implies that local optima are organised into clusters, so that a particular local optimum largely belongs to a particular funnel. Our study is the first to extract and visualise local optima networks for TSP and is based on a sampling methodology relying on the Chained Lin-Kernighan algorithm. We confirm the existence of multiple funnels on two selected TSP instances, finding additional funnels in a previously studied instance. Our results suggests that transitions among funnels are possible using operators such as ‘double-bridge’. However, for consistently escaping sub-optimal funnels, more robust escaping mechanisms are required.

Keywords

Global Optimum Local Optimum Bond Distance Travel Salesman Problem Fitness Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported by the UK’s Engineering and Physical Sciences Research Council [grant number EP/J017515/1].

Data Access. All data generated during this research are openly available from the Zenodo repository (http://doi.org/10.5281/zenodo.20732).

References

  1. 1.
    Applegate, D., Bixby, R., Chvátal, V., Cook, W.: Concorde TSP solver (2003). http://www.math.uwaterloo.ca/tsp/concorde.html
  2. 2.
    Applegate, D., Cook, W., Rohe, A.: Chained Lin-Kernighan for large traveling salesman problems. INFORMS J. Comput. 15, 82–92 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2007)MATHGoogle Scholar
  4. 4.
    Boese, K.D., Kahng, A.B., Muddu, S.: A new adaptive multi-start technique for combinatorial global optimizations. Oper. Res. Lett. 16, 101–113 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal Complex System, 1695 (2006)Google Scholar
  6. 6.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Softw. Pract. Exper. 21(11), 1129–1164 (1991)CrossRefGoogle Scholar
  7. 7.
    Hains, D.R., Whitley, L.D., Howe, A.E.: Revisiting the big valley search space structure in the TSP. J. Oper. Res. Soc. 62(2), 305–312 (2011)CrossRefGoogle Scholar
  8. 8.
    Helsgaun, K.: An effective implementation of the LinKernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iclanzan, D., Daolio, F., Tomassini, M.: Data-driven local optima network characterization of QAPLIB instances. In: Proceedings of the 2014 Conference on Genetic and Evolutionary Computation, GECCO 2014, pp. 453–460. ACM, New York (2014)Google Scholar
  10. 10.
    Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21, 498–516 (1973)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Martin, O., Otto, S.W., Felten, E.W.: Large-step Markov chains for the traveling salesman problem. Complex Syst. 5, 299–326 (1991)MathSciNetMATHGoogle Scholar
  12. 12.
    Möbius, A., Freisleben, B., Merz, P., Schreiber, M.: Combinatorial optimization by iterative partial transcription. Phys. Rev. E 59(4), 4667–4674 (1999)CrossRefGoogle Scholar
  13. 13.
    Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Ochoa, G., Chicano, F., Tinos, R., Whitley, D.: Tunnelling crossover networks. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pp. 449–456. ACM (2015)Google Scholar
  15. 15.
    Ochoa, G., Tomassini, M., Verel, S., Darabos, C.: A study of NK landscapes’ basins and local optima networks. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pp. 555–562. ACM (2008)Google Scholar
  16. 16.
    Ochoa, G., Verel, S., Daolio, F., Tomassini, M.: Local optima networks: a new model of combinatorial fitness landscapes. In: Richter, H., Engelbrecht, A. (eds.) Recent Advances in the Theory and Application of Fitness Landscapes. ECC, vol. 6, pp. 233–262. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  17. 17.
    Reinelt, G.: TSPLIB-a traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991). http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/ MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Verel, S., Ochoa, G., Tomassini, M.: Local optima networks of NK landscapes with neutrality. IEEE Trans. Evol. Comput. 15(6), 783–797 (2011)CrossRefGoogle Scholar
  19. 19.
    Wales, D.J., Miller, M.A., Walsh, T.R.: Archetypal energy landscapes. Nature 394, 758–760 (1998)CrossRefGoogle Scholar
  20. 20.
    Whitley, D., Hains, D., Howe, A.: Tunneling between optima: partition crossover for the traveling salesman problem. In: Proceedings Genetic and Evolutionary Computation Conference, GECCO 2009, pp. 915–922. ACM, New York (2009)Google Scholar
  21. 21.
    Whitley, D., Hains, D., Howe, A.: A hybrid genetic algorithm for the traveling salesman problem using generalized partition crossover. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6238, pp. 566–575. Springer, Heidelberg (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gabriela Ochoa
    • 1
  • Nadarajen Veerapen
    • 1
  • Darrell Whitley
    • 2
  • Edmund K. Burke
    • 1
  1. 1.Computing Science and MathematicsUniversity of StirlingStirlingScotland, UK
  2. 2.Department of Computer ScienceColorado State UniversityFort CollinsUSA

Personalised recommendations