Overview of New Approaches for Approximating TSP

  • Ola Svensson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)

Abstract

In this extended abstract, we survey some of the recent results on approximating the traveling salesman problem on graphic metrics.

We start by briefly explaining the algorithm of Oveis Gharan et al. [1] that has strong similarities to Christofides’ famous 3/2-approximation algorithm. We then explain the main ideas behind an alternative approach introduced by Mömke and the author [2]. The new ingredient in our approach is that it allows for the removal of certain edges while simultaneously yielding a connected, Eulerian graph, which in turn leads to a decreased cost. We also overview the exciting developments for TSP on graphic metrics that rapidly followed: an improved analysis of our algorithm by Mucha [3] yielding an approximation guarantee of 1.44, and the recent developments by Sebö and Vygen [3] who gave a 1.4-approximation algorithm.

Finally, we point out some interesting open problems where our techniques currently fall short of applying to more general metrics.

Keywords

approximation algorithms graph theory traveling salesman problem 

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References

  1. 1.
    Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: [18], pp. 550–559Google Scholar
  2. 2.
    Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: [18], pp. 560–569Google Scholar
  3. 3.
    Mucha, M.: 13/9-approximation for graphic TSP. In: Dürr, C., Wilke, T. (eds.) STACS. LIPIcs, vol. 14, pp. 30–41. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)Google Scholar
  4. 4.
    Lampis, M.: Improved inapproximability for TSP. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 243–253. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University (1976)Google Scholar
  6. 6.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28(4), 1298–1309 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Karloff, H.J. (ed.) SODA, pp. 33–41. ACM/SIAM (1998)Google Scholar
  9. 9.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.H.: An approximation scheme for planar graph TSP. In: Proc. 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp. 640–645. IEEE Computer Society (1995)Google Scholar
  10. 10.
    Goemans, M.X.: Worst-case comparison of valid inequalities for the TSP. Mathematics and Statistics 69(1), 335–349 (1995)MathSciNetMATHGoogle Scholar
  11. 11.
    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: Charikar, M. (ed.) SODA, pp. 379–389. SIAM (2010)Google Scholar
  12. 12.
    Frederickson, G.N., Jájá, J.: On the relationship between the biconnectivity augmentation and travelling salesman problems. Theoretical Computer Science 19(2), 189–201 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Monma, C.L., Munson, B.S., Pulleyblank, W.R.: Minimum-weight two-connected spanning networks. Mathematical Programming 46(1), 153–171 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Boyd, S., Sitters, R., van der Ster, S., Stougie, L.: TSP on cubic and subcubic graphs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 65–77. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. Operations Research Letters 33(5), 467–474 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. CoRR abs/1201.1870 (2012)Google Scholar
  17. 17.
    Vygen, J.: New approximation algorithms for the TSP. In: OPTIMA, vol. 90, http://www.mathopt.org/Optima-Issues/optima90.pdf
  18. 18.
    Ostrovsky, R. (ed.): IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25. IEEE (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ola Svensson
    • 1
  1. 1.École Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

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