Swarm Intelligence

, Volume 6, Issue 1, pp 1–21 | Cite as

Theoretical analysis of two ACO approaches for the traveling salesman problem

  • Timo Kötzing
  • Frank Neumann
  • Heiko Röglin
  • Carsten Witt
Article

Abstract

Bioinspired algorithms, such as evolutionary algorithms and ant colony optimization, are widely used for different combinatorial optimization problems. These algorithms rely heavily on the use of randomness and are hard to understand from a theoretical point of view. This paper contributes to the theoretical analysis of ant colony optimization and studies this type of algorithm on one of the most prominent combinatorial optimization problems, namely the traveling salesperson problem (TSP). We present a new construction graph and show that it has a stronger local property than one commonly used for constructing solutions of the TSP. The rigorous runtime analysis for two ant colony optimization algorithms, based on these two construction procedures, shows that they lead to good approximation in expected polynomial time on random instances. Furthermore, we point out in which situations our algorithms get trapped in local optima and show where the use of the right amount of heuristic information is provably beneficial.

Keywords

Ant colony optimization Traveling salesperson problem Run time analysis Approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chandra, B., Karloff, H. J., & Tovey, C. A. (1999). New results on the old k-Opt algorithm for the traveling salesman problem. SIAM Journal on Computing, 28(6), 1998–2029. CrossRefMATHMathSciNetGoogle Scholar
  2. Dorigo, M., & Gambardella, L. M. (1997). Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation, 1(1), 53–66. CrossRefGoogle Scholar
  3. Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press. CrossRefMATHGoogle Scholar
  4. Eiben, A., & Smith, J. (2007). Introduction to evolutionary computing (2nd ed.). Berlin: Springer. Google Scholar
  5. Englert, M., Röglin, H., & Vöcking, B. (2007). Worst case and probabilistic analysis of the 2-opt algorithm for the TSP: extended abstract. In N. Bansal, K. Pruhs, & C. Stein (Eds.), SODA’07: Proceedings of the eighteenth annual ACM–SIAM symposium on discrete algorithms (pp. 1295–1304). Philadelphia: Society for Industrial and Applied Mathematics. Google Scholar
  6. Gutjahr, W. J. (2007). Mathematical runtime analysis of ACO algorithms: survey on an emerging issue. Swarm Intelligence, 1(1), 59–79. CrossRefGoogle Scholar
  7. Gutjahr, W. J., & Sebastiani, G. (2008). Runtime analysis of ant colony optimization with best-so-far reinforcement. Methodology and Computing in Applied Probability, 10(3), 409–433. CrossRefMATHMathSciNetGoogle Scholar
  8. He, J., & Yao, X. (2004). A study of drift analysis for estimating computation time of evolutionary algorithms. Natural Computing, 3(1), 21–35. CrossRefMATHMathSciNetGoogle Scholar
  9. Horoba, C., & Sudholt, D. (2009). Running time analysis of ACO systems for shortest path problems. In T. Stützle, M. Birattari, & H. H. Hoos (Eds.), Lecture notes in computer science: Vol. 5752. Engineering stochastic local search algorithms. Designing, implementing and analyzing effective heuristics, second international workshop, SLS, 2009 (pp. 76–91). Berlin: Springer. CrossRefGoogle Scholar
  10. Johnson, D. S., & McGeoch, L. A. (1997). The traveling salesman problem: a case study in local optimization. In E. H. L. Aarts & J. K. Lenstra (Eds.), Local search in combinatorial optimization. Somerset: Wiley. Google Scholar
  11. Kötzing, T., Neumann, F., Röglin, H., & Witt, C. (2010). Theoretical properties of two ACO approaches for the traveling salesman problem. In M. Dorigo, M. Birattari, G. A. D. Caro, R. Doursat, A. P. Engelbrecht, D. Floreano, L. M. Gambardella, R. Groß, E. Sahin, H. Sayama, & T. Stützle (Eds.), Lecture notes in computer science: Vol. 6234. Swarm intelligence, 7th international conference, ANTS, 2010 (pp. 324–335). Berlin: Springer. Google Scholar
  12. Neumann, F., Sudholt, D., & Witt, C. (2008). Rigorous analyses for the combination of ant colony optimization and local search. In M. Dorigo, M. Birattari, C. Blum, M. Clerc, T. Stützle, & A. F. T. Winfield (Eds.), Lecture notes in computer science: Vol. 5217. Ant colony optimization and swarm intelligence, 6th international conference, ANTS, 2008 (pp. 132–143). Berlin: Springer. CrossRefGoogle Scholar
  13. Neumann, F., Sudholt, D., & Witt, C. (2009). Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intelligence, 3(1), 35–68. CrossRefGoogle Scholar
  14. Neumann, F., & Witt, C. (2009). Runtime analysis of a simple ant colony optimization algorithm. Algorithmica, 54(2), 243–255. CrossRefMATHMathSciNetGoogle Scholar
  15. Neumann, F., & Witt, C. (2010). Ant colony optimization and the minimum spanning tree problem. Theoretical Computer Science, 411(25), 2406–2413. CrossRefMATHMathSciNetGoogle Scholar
  16. Neumann, F., & Witt, C. (2010). Bioinspired computation in combinatorial optimization—algorithms and their computational complexity. Berlin: Springer. CrossRefMATHGoogle Scholar
  17. Spielman, D. A., & Teng, S.-H. (2004). Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51(3), 385–463. CrossRefMATHMathSciNetGoogle Scholar
  18. Stützle, T., & Hoos, H. H. (2000). \(\mathcal{MAX}\)\(\mathcal{MIN}\) ant system. Future Generations Computer Systems, 16(8), 889–914. CrossRefGoogle Scholar
  19. Zhou, Y. (2009). Runtime analysis of an ant colony optimization algorithm for TSP instances. IEEE Transactions on Evolutionary Computation, 13(5), 1083–1092. CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2011

Authors and Affiliations

  • Timo Kötzing
    • 1
  • Frank Neumann
    • 2
  • Heiko Röglin
    • 3
  • Carsten Witt
    • 4
  1. 1.Algorithms and Complexity, Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.School of Computer ScienceUniversity of AdelaideAdelaideAustralia
  3. 3.Department of Computer ScienceUniversity of BonnBonnGermany
  4. 4.DTU InformaticsTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations