Improved Approximations for TSP with Simple Precedence Constraints

(Extended Abstract)
  • Hans-Joachim Böckenhauer
  • Ralf Klasing
  • Tobias Mömke
  • Monika Steinová
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)

Abstract

In this paper, we consider variants of the traveling salesman problem with precedence constraints. We characterize hard input instances for Christofides’ algorithm and Hoogeveen’s algorithm by relating the two underlying problems, i. e., the traveling salesman problem and the problem of finding a minimum-weight Hamiltonian path between two prespecified vertices. We show that the sets of metric worst-case instances for both algorithms are disjoint in the following sense. There is an algorithm that, for any input instance, either finds a Hamiltonian tour that is significantly better than 1.5-approximative or a set of Hamiltonian paths between all pairs of endpoints, all of which are significantly better than 5/3-approximative.

In the second part of the paper, we give improved algorithms for the ordered TSP, i. e., the TSP, where the precedence constraints are such that a given subset of vertices has to be visited in some prescribed linear order. For the metric case, we present an algorithm that guarantees an approximation ratio of 2.5 − 2/k, where k is the number of ordered vertices. For near-metric input instances satisfying a β-relaxed triangle inequality, we improve the best previously known ratio to \(k\beta^{\log_2 (3k-3)}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andreae, T.: On the traveling salesman problem restricted to inputs satisfying a relaxed triangle inequality. Networks 38(2), 59–67 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bandelt, H.J., Crama, Y., Spieksma, F.C.R.: Approximation algorithms for multi-dimensional assignment problems with decomposable costs. Discrete Appl. Math. 49(1-3), 25–50 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Böckenhauer, H.-J., Hromkovič, J., Kneis, J., Kupke, J.: On the approximation hardness of some generalizations of TSP (extended abstract). In: Arge, L., Freivalds, R.V. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 184–195. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Tech. Rep. 388, Graduate School of Industrial Administration, Carnegie-Mellon University (1976)Google Scholar
  5. 5.
    Eppstein, D.: Paired approximation problems and incompatible inapproximabilities. In: Charikar, M. (ed.) Proc. of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 1076–1086. SIAM, New York (2010)CrossRefGoogle Scholar
  6. 6.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Gutin, G., Punnen, A.P. (eds.): The Traveling Salesman Problem and Its Variations. Combinatorial Optimization. Springer, New York (2007)MATHGoogle Scholar
  8. 8.
    Guttmann-Beck, N., Hassin, R., Khuller, S., Raghavachari, B.: Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28(4), 422–437 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10(5), 291–295 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sahni, S., Gonzalez, T.F.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976)MathSciNetMATHGoogle Scholar
  11. 11.
    Vassilevska, V., Williams, R., Woo, S.L.M.: Confronting hardness using a hybrid approach. In: Proc. of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 1–10. SIAM, New York (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Ralf Klasing
    • 2
  • Tobias Mömke
    • 1
  • Monika Steinová
    • 1
  1. 1.Department of Computer ScienceETH ZurichSwitzerland
  2. 2.CNRS - LaBRI - Université Bordeaux 1Talence cedexFrance

Personalised recommendations