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Optimal TSP tour length estimation using Sammon maps

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Abstract

The optimal tour length of a non-Euclidean traveling salesman problem (TSP) can be estimated using the locations of vertices and the circuity factor. In this paper, we propose a method to estimate the optimal tour length of a non-Euclidean TSP using Sammon mapping. While providing accuracy comparable to the approach using the circuity factor, this new method has a number of advantages.

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Data availibility statement

The datasets generated during and/or analysed as part of the current study are available from the corresponding author upon request.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, USA

    Shuhan Kou

  2. Robert H. Smith School of Business, University of Maryland, College Park, MD, USA

    Bruce Golden

  3. Business School, University of Colorado, Denver, CO, USA

    Stefan Poikonen

Authors
  1. Shuhan Kou
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  2. Bruce Golden
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  3. Stefan Poikonen
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Correspondence to Shuhan Kou.

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Appendix

Appendix

For the 1 km\(^2\) São Paulo road map instances in Fig. 4, after obtaining the coefficients of the \(\sqrt{NA_{Sammon}}\) model using linear regression, we also generate the two residual plots in Fig. 14 below for two variables in the predictor set: number of vertices, N, and the convex hull area based on projected vertex coordinates in a Sammon map, \(A_{Sammon}\).

Fig. 14
figure 14

Residual plots of the regression on 1 km\(^2\) São Paulo road maps

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There is no obvious pattern in the residual plot with respect to \(A_{Sammon}\), and the value of the residual does not seem to depend on the value of \(A_{Sammon}\). For the residual plot with respect to N, one may find some degree of heteroskedasticity, and question if an increase in N will decrease the predictive ability of the \(\sqrt{NA_{Sammon}}\) predictor. To address this concern, we include the residual plot for the variable N for the 4 km\(^2\) São Paulo road map instances in Fig. 15, and we observe that it is not the case that the residual grows as N increases.

Fig. 15
figure 15

Residual plot of the regression on 4 km\(^2\) São Paulo road maps

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Kou, S., Golden, B. & Poikonen, S. Optimal TSP tour length estimation using Sammon maps. Optim Lett 17, 89–105 (2023). https://doi.org/10.1007/s11590-022-01937-y

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