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


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.


TSP Empirical complexity Exact algorithms 



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)


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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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