Advertisement

Edge Elimination in TSP Instances

  • Stefan HougardyEmail author
  • Rasmus T. Schroeder
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

The Traveling Salesman Problem is one of the best studied NP-hard problems in combinatorial optimization. Powerful methods have been developed over the last 60 years to find optimum solutions to large TSP instances. The largest TSP instance so far that has been solved optimally has 85,900 vertices. Its solution required more than 136 years of total CPU time using the branch-and-cut based Concorde TSP code [1]. In this paper we present graph theoretic results that allow to prove that some edges of a TSP instance cannot occur in any optimum TSP tour. Based on these results we propose a combinatorial algorithm to identify such edges. The runtime of the main part of our algorithm is \(O(n^2 \log n)\) for an \(n\)-vertex TSP instance. By combining our approach with the Concorde TSP solver we are able to solve a large TSPLIB instance more than 11 times faster than Concorde alone.

Keywords

Traveling salesman problem Exact algorithm 

Notes

Acknowledgement

We are very grateful to Bill Cook for supplying us with some data and several helpful comments. We also thank our reviewers for their careful reading and useful comments.

References

  1. 1.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)Google Scholar
  2. 2.
    W.J. Cook’s TSP website at http://www.math.uwaterloo.ca/tsp/
  3. 3.
    W.J. Cook. Personal communication, December 2013Google Scholar
  4. 4.
    Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. J. Oper. Res. Soc. Am. 2(4), 393–410 (1954)MathSciNetGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Helsgaun, K.: General \(k\)-opt submoves for the Lin-Kernighan TSP heuristic. Math. Program. Comput. 1(2–3), 119–163 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hougardy, S., Schroeder, R.: Edge Elimination in TSP Instances (2014). arXiv:1402.7301v1
  9. 9.
  10. 10.
    Jonker, R., Volgenant, T.: Nonoptimal edges for the symmetric traveling salesman problem. Oper. Res. 32(4), 837–846 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Reinelt, G.: TSPLIB 95. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Heidelberg (1995)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations