Mathematical Programming

, Volume 172, Issue 1–2, pp 399–413 | Cite as

Improved approximations for cubic bipartite and cubic TSP

  • Anke van ZuylenEmail author
Full Length Paper Series B


We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs. For connected cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi 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 can be combined with the techniques of Correa, Larré and Soto, to obtain a tour of length at most \((4/3-1/8754)n\).


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

Mathematics Subject Classification

68W25 approximation algorithms 68W40 analysis of algorithms 05C85 graph algorithms 



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.

Supplementary material


  1. 1.
    Aggarwal, N., Garg, N., Gupta, S.: A 4/3-approximation for TSP on cubic 3-edge-connected graphs. (2011)
  2. 2.
    Barnette, D.: Conjecture 5. In: Tutte, W.T. (ed.) Recent progress in combinatorics: proceedings of the third waterloo conference on combinatorics, May 1968. Academic Press, New York (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)MathSciNetCrossRefGoogle 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 (1976)Google Scholar
  6. 6.
    Correa, J., Larré, O., Soto, J.: TSP tours in cubic graphs: beyond 4/3. SIAM J. Discrete Math. 29(2), 915–939 (2015) (Preliminary version appeared in ESA 2012: 790-801)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Karp, J., Ravi, R.: A 9/7-approximation algorithm for graphic TSP in cubic bipartite graphs. Discrete Appl. Math. 209, 164–216 (2016) (Preliminary version appeared in (APPROX-RANDOM 2014): pp. 284–296, 2014)Google Scholar
  11. 11.
    Mömke, T., Svensson, O.: Removing and adding edges for the traveling salesman problem. J. ACM 63(1), 2 (2016) (Preliminary version appeared in FOCS 2011: pp. 560–569, 2011)Google Scholar
  12. 12.
    Mucha, M.: 13/9 -approximation for graphic TSP. Theory Comput. Syst. 55(4), 640–657 (2014) (Preliminary version appeared in STACS 2012: 30–41, 2012)Google Scholar
  13. 13.
    Petersen, J.: Die Theorie der regulären graphs. Acta Math. 15, 193–220 (1891)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations