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

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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.

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Notes

  1. We further discuss the reasons for holding such perceptions in Sect. 4.

  2. Note that on larger instances, no optimal solutions were found in Concorde’s initial CLK phase.

  3. We note that for solving very large TSP instances, non-standard settings of Concorde are used, and special techniques are exploited to obtain feasible computation times on clusters of parallel machines; for details, see Chapter 16 of [3]. In fact, for the proof of optimality for the instance sw24978 (which involves the locations of cities in Sweden), it was stated explicitly that “the long Sweden computation benefited greatly form the accurate upper bounds provided by Helsgaun’s tour” [3], p. 517.

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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.

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Authors and Affiliations

  1. Computer Science Department, University of British Columbia, Vancouver, BC, Canada

    Holger H. Hoos

  2. IRIDIA, CoDE, Université Libre de Bruxelles (ULB), Brussels, Belgium

    Thomas Stützle

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  1. Holger H. Hoos
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  2. Thomas Stützle
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Correspondence to Holger H. Hoos.

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Hoos, H.H., Stützle, T. On the empirical time complexity of finding optimal solutions vs proving optimality for Euclidean TSP instances. Optim Lett 9, 1247–1254 (2015). https://doi.org/10.1007/s11590-014-0828-5

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