# Improved approximations for cubic bipartite and cubic TSP

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## Abstract

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 \((4/3-1/8754)n\).

## Keywords

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## 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.

## Supplementary material

## References

- 1.Aggarwal, N., Garg, N., Gupta, S.: A 4/3-approximation for TSP on cubic 3-edge-connected graphs. http://arxiv.org/abs/1101.5586 (2011)
- 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.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.Candráková, B., Lukotka, R.: Cubic TSP—a 1.3-approximation. CoRR, abs/1506.06369 (2015)Google Scholar
- 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.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.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.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.Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res.
**18**, 1138–1162 (1970)MathSciNetCrossRefGoogle Scholar - 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.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.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.Petersen, J.: Die Theorie der regulären graphs. Acta Math.
**15**, 193–220 (1891)MathSciNetCrossRefGoogle Scholar - 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