Advertisement

Improved Approximations for Cubic Bipartite and Cubic TSP

  • Anke van Zuylen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)

Abstract

We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs. For cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi [10] by giving a “local improvement” algorithm that finds a tour of length at most \(5/4n-2\). For 2-connected cubic graphs, we show that the techniques of Mömke and Svensson [11] can be combined with the techniques of Correa et al. [6], to obtain a tour of length at most \((4/3-1/8754)n\).

Keywords

Traveling salesman problem Approximation algorithm Cubic bipartite graphs Cubic graphs Barnette’s conjecture 

Notes

Acknowledgements

The author would like to thank Marcin Mucha for careful reading and pointing out an omission in a previous version, Frans Schalekamp for helpful discussions, and an anonymous reviewer for suggesting the simplified proof for the result in Sect. 3 for cubic non-bipartite graphs. Other anonymous reviewers are acknowledged for helpful feedback on the presentation of the algorithm for bipartite cubic graphs.

References

  1. 1.
    Aggarwal, N., Garg, N., Gupta, S.: A 4/3-approximation for TSP on cubic 3-edge-connected graphs (2011). http://arxiv.org/abs/1101.5586
  2. 2.
    Barnette, D.W.: Conjecture 5. In: Recent Progress in Combinatorics (1969)Google Scholar
  3. 3.
    Boyd, S., Sitters, R., van der Ster, S., Stougie, L.: The traveling salesman problem on cubic and subcubic graphs. Math. Program. 144(1–2), 227–245 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Candráková, B., Lukotka, R.: Cubic TSP - a 1.3-approximation. CoRR abs/1506.06369 (2015)Google Scholar
  5. 5.
    Christofides, N.: Worst case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA (1976)Google Scholar
  6. 6.
    Correa, J.R., Larré, O., Soto, J.A.: TSP tours in cubic graphs: beyond 4/3. SIAM J. Discrete Math. 29(2), 915–939 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dantzig, G.B., Fulkerson, D.R., Johnson, S.M.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)MathSciNetGoogle Scholar
  8. 8.
    Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. Oper. Res. Lett. 33(5), 467–474 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karp, J., Ravi, R.: A 9/7-approximation algorithm for graphic TSP in cubic bipartite graphs. In: Approximation, Randomization, and Combinatorial Optimization (APPROX-RANDOM). LIPIcs, vol. 28, pp. 284–296. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  11. 11.
    Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: Proceedings of the 52th Annual Symposium on Foundations of Computer Science, pp. 560–569 (2011)Google Scholar
  12. 12.
    Mucha, M.: 13/9-approximation for graphic TSP. Theory Comput. Syst. 55(4), 640–657 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    van Zuylen, A.: Improved approximations for cubic and cubic bipartite TSP. CoRR abs/1507.07121 (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsCollege of William & MaryWilliamsburgUSA

Personalised recommendations