Reoptimization of Traveling Salesperson Problems: Changing Single Edge-Weights

  • Tobias Berg
  • Harald Hempel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5457)

Abstract

We consider the following optimization problem: Given an instance of an optimization problem and some optimum solution for this instance, we want to find a good solution for a slightly modified instance. Additionally, the scenario is addressed where the solution for the original instance is not an arbitrary optimum solution, but is chosen amongst all optimum solutions in a most helpful way. In this context, we examine reoptimization of the travelling salesperson problem, in particular MinTSP and MaxTSP as well as their corresponding metric versions. We study the case where the weight of a single edge is modified. Our main results are the following: existence of a 4/3-approximation for the metric MinTSP-problem, a 5/4-approximation for MaxTSP, and a PTAS for the metric version of MaxTSP.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem, Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976)Google Scholar
  2. 2.
    Serdyukov, A.I.: An algorithm with an estimate for the traveling salesman problem of the maximum. Upravlyaemye Sistemy 25, 80–86 (1984)MathSciNetMATHGoogle Scholar
  3. 3.
    Chen, Z.Z., Nagoya, T.: Improved approximation algorithms for metric MaxTSP. J. Comb. Optim. 13(4), 321–336 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ausiello, G., Escoffier, B., Monnot, J., Paschos, V.T.: Reoptimization of minimum and maximum traveling salesman’s tours. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 196–207. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Böckenhauer, H.J., Hromkovič, J., Mömke, T., Widmayer, P.: On the hardness of reoptimization. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 50–65. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Liberatore, P.: The complexity of modified instances. arXiv.org cs/0402053 (2004)Google Scholar
  7. 7.
    Böckenhauer, H.J., Forlizzi, L., Hromkovič, J., Kneis, J., Kupke, J., Proietti, G., Widmayer, P.: On the approximability of TSP on local modifications of optimally solved instances. Algorithmic Oper. Res. 2(2), 83–93 (2007)MathSciNetMATHGoogle Scholar
  8. 8.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial optimization: algorithms and complexity. Dover Publications Inc., Mineola (1998); corrected reprint of the 1982 originalMATHGoogle Scholar
  9. 9.
    Lovász, L., Plummer, M.: Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland, Amsterdam (1986)MATHGoogle Scholar
  10. 10.
    Gabow, H.: Implementation of algorithms for maximum matching on nonbipartite graphs, Ph.D. Thesis, Stanford University (1974)Google Scholar
  11. 11.
    Barvinok, A., Gimadi, E.K., Serdyukow, A.I.: The Maximum Traveling Salesman Problem. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and Its Variations, pp. 585–608. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Tobias Berg
    • 1
  • Harald Hempel
    • 1
  1. 1.Fakultät für Mathematik und InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations