Constant Factor Approximation for ATSP with Two Edge Weights

(Extended Abstract)
  • Ola Svensson
  • Jakub Tarnawski
  • László A. Végh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)

Abstract

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently in [17]. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

References

  1. 1.
    Anari, N., Gharan, S.O.: Effective-resistance-reducing flows and asymmetric TSP. CoRR, abs/1411.4613 (2014)Google Scholar
  2. 2.
    Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of SODA, vol. 98, pp. 33–41 (1998)Google Scholar
  3. 3.
    Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of SODA, pp. 379–389 (2010)Google Scholar
  4. 4.
    Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1, 2)-TSP. In: Proceedings of SODA, pp. 641–648 (2006)Google Scholar
  5. 5.
    Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 61–71. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, DTIC Document (1976)Google Scholar
  7. 7.
    Erickson, J., Sidiropoulos, A.: A near-optimal approximation algorithm for asymmetric TSP on embedded graphs. In: Proceedings of SOCG, p. 130 (2014)Google Scholar
  8. 8.
    Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gharan, S.O., Saberi, A.: The asymmetric traveling salesman problem on graphs with bounded genus. In: Proceedings of SODA, pp. 967–975. SIAM (2011)Google Scholar
  10. 10.
    Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of FOCS, pp. 550–559 (2011)Google Scholar
  11. 11.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.H.: An approximation scheme for planar graph TSP. In: Proceedings of FOCS, pp. 640–645 (1995)Google Scholar
  12. 12.
    Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. J. Comput. Syst. Sci. 81(8), 1665–1677 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: 2011 Proceedings of FOCS, pp. 560–569 (2011)Google Scholar
  14. 14.
    Mucha, M.: 13/9-approximation for graphic TSP. In: Proceedings of STACS, pp. 30–41 (2012)Google Scholar
  15. 15.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18(1), 1–11 (1993)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica 34(5), 597–629 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Svensson, O.: Approximating ATSP by relaxing connectivity. In: Proceedings of FOCS (2015)Google Scholar
  18. 18.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, New York (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ola Svensson
    • 1
  • Jakub Tarnawski
    • 1
  • László A. Végh
    • 2
  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.London School of EconomicsLondonUK

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