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Better s-t-Tours by Gao Trees

  • Corinna GottschalkEmail author
  • Jens Vygen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)

Abstract

We consider the s-t-path TSP: given a finite metric space with two elements s and t, we look for a path from s to t that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution \(x^*\) as a convex combination of spanning trees. Gao showed that there exists a spanning tree in the support of \(x^*\) that has only one edge in each narrow cut (i.e., each cut C with \(x^*(C)<2\)). Our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such “Gao trees” simultaneously at all sufficiently narrow cuts.

Keywords

Narrow Cuts Approximation Ratio Main Theorem Says Christofides Algorithm Eulerian Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

We thank Kanstantsin Pashkovich for allowing us to include his idea described in the second half of Sect. 3.

References

  1. An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the \(s\)-\(t\) path TSP. J. ACM 62, Article 34, 34:1–34:28 (2015)Google Scholar
  2. Asadpour, A., Goemans, M.X., Mądry, A., Oveis Gharan, S., Saberi, A.: An \(O(\log n/ \log \log n)\)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 379–389 (2010)Google Scholar
  3. Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976)Google Scholar
  4. Edmonds, J.: The Chinese postman’s problem. Bull. Oper. Res. Soc. Am. 13, B-73 (1965)MathSciNetGoogle Scholar
  5. Edmonds, J., Johnson, E.L.: Matching, Euler tours and the Chinese postman. Math. Program. 5, 88–124 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Gao, Z.: An LP-based \(\frac{3}{2}\)-approximation algorithm for the \(s\)-\(t\) path graph traveling salesman problem. Oper. Res. Lett. 41, 615–617 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gao, Z.: On the metric \(s\)-\(t\) path traveling salesman problem. SIAM J. Discrete Math. 29, 1133–1149 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Genova, K., Williamson, D.P.: An experimental evaluation of the best-of-many Christofides’ algorithm for the traveling salesman problem. In: Bansal, N., Finocchi, I. (eds.) Algorithms – ESA 2015. LNCS, pp. 570–581. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  9. Goemans, M.X.: Minimum bounded-degree spanning trees. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 273–282 (2006)Google Scholar
  10. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10, 291–295 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011), pp. 550–559 (2011)Google Scholar
  14. Sebő, A.: Eight-fifth approximation for the path TSP. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 362–374. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. Sebö, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica 34, 597–629 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Vygen, J.: New approximation algorithms for the TSP. OPTIMA 90, 1–12 (2012)Google Scholar
  17. Vygen, J.: Reassembling trees for the traveling salesman. SIAM J. Discrete Math. To appear. arXiv:1502.03715

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany
  2. 2.University of BonnBonnGermany

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