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Optimization Letters

, Volume 9, Issue 6, pp 1247–1254 | Cite as

On the empirical time complexity of finding optimal solutions vs proving optimality for Euclidean TSP instances

  • Holger H. Hoos
  • Thomas Stützle
Short Communication

Abstract

We investigate the empirical performance of the long-standing state-of-the-art exact TSP solver Concorde on various classes of Euclidean TSP instances and show that, surprisingly, the time spent until the first optimal solution is found accounts for a large fraction of Concorde’s overall running time. This finding holds for the widely studied random uniform Euclidean (RUE) instances as well as for several other widely studied sets of Euclidean TSP instances. On RUE instances, the median fraction of Concorde’s total running time spent until an optimal solution is found ranges from 0.77 for \(n=500\) to 0.97 for \(n=3{,}500\); on TSPLIB, National and VLSI instances, we pegged it at 0.86, 0.74 and 0.61, respectively, with a tendency of even smaller values for larger instances.

Keywords

TSP Empirical complexity Exact algorithms 

Notes

Acknowledgments

We gratefully acknowledge helpful input from David Mitchell on connections with complexity theory. Furthermore, we thank the anonymous reviewers for their useful comments. This work was supported by the COMEX project within the Interuniversity Attraction Poles Programme of the Belgian Science Policy Office. H. H. acknowledges support through an NSERC Discovery Grant. T. S. acknowledges support from the Belgian F. R. S.-FNRS, of which he is a senior research associate.

Supplementary material

11590_2014_828_MOESM1_ESM.pdf (163 kb)
Supplementary material 1 (pdf 164 KB)

References

  1. 1.
    The traveling salesman problem. Version visited last on 15 April 2014. http://www.math.uwaterloo.ca/tsp/ (2014)
  2. 2.
    Applegate, D.: Personal communication (2009)Google Scholar
  3. 3.
    Applegate, D., Bixby, R.E., Chvatal, V., Cook, W.J.: The traveling salesman problem: a computational study. Princeton University Press, Princeton (2006)Google Scholar
  4. 4.
    Applegate, D., Bixby, R.E., Chvatal, V., Cook, W.J.: Concorde TSP solver. Version visited last on 15 April 2014. http://www.math.uwaterloo.ca/tsp/concorde.html (2014)
  5. 5.
    Beame, P., Pitassi, T.: Propositional proof complexity: past, present, and future. In: Current trends in theoretical computer science: entering the 21st century, pp. 42–70. World Scientific Publishing (2001)Google Scholar
  6. 6.
    Dash, S.: Exponential lower bounds on the lengths of some classes of branch-and-cut proofs. Math. Oper. Res. 30(3), 678–700 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126, 106–130 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Helsgaun, K.: General k-opt submoves for the Lin-Kernighan TSP heuristic. Math. Program. Comput. 1(2–3), 119–163 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoos, H.H., Stützle, T.: On the empirical scaling of run-time for finding optimal solutions to the traveling salesman problem. Eur. J. Oper. Res. 238(1), 87–94 (2014)CrossRefGoogle Scholar
  10. 10.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan., A.H.G, Shmoys, D.B.: The traveling salesman problem. Wiley, Chichester (1985)Google Scholar
  11. 11.
    Nagata, Y., Kobayashi, S.: A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem. INFORMS J. Comput. 25(2), 346–363 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175, 512–525 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Reinelt, G.: The traveling salesman: computational solutions for TSP applications. Lecture Notes in Computer Science, vol. 840. Springer, Heidelberg, Germany (1994)Google Scholar
  14. 14.
    Reinelt, G.: TSPLIB. Version visited last on 15 June 2012, http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95 (2012)

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of British ColumbiaVancouverCanada
  2. 2.IRIDIA, CoDEUniversité Libre de Bruxelles (ULB)BrusselsBelgium

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