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Computing the variance of tour costs over the solution space of the TSP in polynomial time

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Abstract

We give an O(n 2) time algorithm to find the population variance of tour costs over the solution space of the n city symmetric Traveling Salesman Problem (TSP). The algorithm has application in both the stochastic case, where the problem is specified in terms of edge costs which are pairwise independently distributed random variables with known mean and variance, and the numeric edge cost case.

We apply this result to provide empirical evidence that, in a range of real world problem sets, the optimal tour cost correlates with a simple function of the mean and variance of tour costs.

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Authors and Affiliations

  1. Faculty of Engineering and Information Technology, University of Technology, Sydney, Sydney, Australia

    Paul J. Sutcliffe, Andrew Solomon & Jenny Edwards

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  1. Paul J. Sutcliffe
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  2. Andrew Solomon
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  3. Jenny Edwards
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Correspondence to Paul J. Sutcliffe.

Appendix: The mean and variance of TSPLIB problems

Appendix: The mean and variance of TSPLIB problems

Tables 5 and 6 provide the mean and variance of tour costs over the solution space of 107 instances from the well known TSPLIB [24] archive.

Table 5 The mean, μ and variance, σ 2 of TSPLIB problems instances a280 to pla3797
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Table 6 The mean, μ and variance, σ 2 of TSPLIB problems instances pr to v
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Sutcliffe, P.J., Solomon, A. & Edwards, J. Computing the variance of tour costs over the solution space of the TSP in polynomial time. Comput Optim Appl 53, 711–728 (2012). https://doi.org/10.1007/s10589-012-9472-0

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