Theoretical Properties of Two ACO Approaches for the Traveling Salesman Problem

  • Timo Kötzing
  • Frank Neumann
  • Heiko Röglin
  • Carsten Witt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6234)

Abstract

Ant colony optimization (ACO) has been widely used for different combinatorial optimization problems. In this paper, we investigate ACO algorithms with respect to their runtime behavior for the traveling salesperson (TSP) problem. We present a new construction graph and show that it has a stronger local property than the given input graph which is often used for constructing solutions. Later on, we investigate ACO algorithms for both construction graphs on random instances and show that they achieve a good approximation in expected polynomial time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambrigde (2004)MATHGoogle Scholar
  3. 3.
    Eiben, A., Smith, J.: Introduction to Evolutionary Computing, 2nd edn. Springer, Berlin (2007)Google Scholar
  4. 4.
    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, Philadelphia (2007)Google Scholar
  5. 5.
    Gutjahr, W.J., Sebastiani, G.: Runtime analysis of ant colony optimization with best-so-far reinforcement. Methodology and Computing in Applied Probability 10, 409–433 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Horoba, C., Sudholt, D.: Running time analysis of ACO systems for shortest path problems. In: Stützle, T., Birattari, M., Hoos, H.H. (eds.) SLS 2009. LNCS, vol. 5752, pp. 76–91. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    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, Chichester (1997)Google Scholar
  8. 8.
    Neumann, F., Sudholt, D., Witt, C.: Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intelligence 3(1), 35–68 (2009)CrossRefGoogle Scholar
  9. 9.
    Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. Algorithmica 54(2), 243–255 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Spielman, D.A., Teng, S.H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)MATHMathSciNetGoogle Scholar
  11. 11.
    Zhou, Y.: Runtime analysis of an ant colony optimization algorithm for TSP instances. IEEE Transactions on Evolutionary Computation 13(5), 1083–1092 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Timo Kötzing
    • 1
  • Frank Neumann
    • 1
  • Heiko Röglin
    • 2
  • Carsten Witt
    • 3
  1. 1.Max-Planck-Institut für InformatikAlgorithms and ComplexitySaarbrückenGermany
  2. 2.Department of Quantitative EconomicsMaastricht UniversityThe Netherlands
  3. 3.DTU InformaticsTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations