On the empirical scaling of running time for finding optimal solutions to the TSP

Abstract

We study the empirical scaling of the running time required by state-of-the-art exact and inexact TSP algorithms for finding optimal solutions to Euclidean TSP instances as a function of instance size. In particular, we use a recently introduced statistical approach to obtain scaling models from observed performance data and to assess the accuracy of these models. For Concorde, the long-standing state-of-the-art exact TSP solver, we compare the scaling of the running time until an optimal solution is first encountered (the finding time) and that of the overall running time, which adds to the finding time the additional time needed to complete the proof of optimality. For two state-of-the-art inexact TSP solvers, LKH and EAX, we compare the scaling of their running time for finding an optimal solution to a given instance; we also compare the resulting models to that for the scaling of Concorde’s finding time, presenting evidence that both inexact TSP solvers show significantly better scaling behaviour than Concorde.

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Notes

  1. 1.

    Note that the running times of Concorde follow a probability distribution, as different search paths are taken for different pseudo-random number seeds. This variability in running time may be exploited through multiple parallel runs.

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Acknowledgements

This work received support from the COMEX project within the Interuniversity Attraction Poles Programme of the Belgian Science Policy Office. Thomas Stützle acknowledges support from the Belgian F.R.S.-FNRS, of which he is a research director. Holger Hoos acknowledges support through an NSERC Discovery Grant and a computing resource allocation by Compute Canada / Calcul Canada.

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Correspondence to Holger H. Hoos or Thomas Stützle.

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Mu, Z., Dubois-Lacoste, J., Hoos, H.H. et al. On the empirical scaling of running time for finding optimal solutions to the TSP. J Heuristics 24, 879–898 (2018). https://doi.org/10.1007/s10732-018-9374-0

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Keywords

  • Traveling salesman problem
  • Lin-Kernighan algorithm
  • Evolutionary algorithms
  • Scaling analysis
  • Concorde
  • Empirical analysis