# Traveling salesman problem across well-connected cities and with location-dependent edge lengths

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## Abstract

Consider \(n\) nodes \(\{X_i\}_{1 \le i \le n}\) distributed independently across \(N\) cities located in the unit square \(S\), each according to a certain distribution \(g_N(\cdot )\). Each city is modelled as an \(r_n \times r_n\) square and \({\mathrm{TSP}}C_n\) denotes the weight of the minimum weighted length cycle containing all the \(n\) nodes, where the edge length between nodes \(X_i\) and \(X_j\) is location-dependent and based on a metric \(d\) that is equivalent to the Euclidean metric. We obtain variance estimates for \({\mathrm{TSP}}C_n\) and prove that if the cities are well-connected in a certain sense, then \({\mathrm{TSP}}C_n\) appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for \({\mathrm{TSP}}C_n\). Using the proof techniques, we also study results \({\mathrm{TSP}}_n\) of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square \(S\) with location-dependent edge lengths. We obtain variance estimates and convergence in probability for \({\mathrm{TSP}}_n\) appropriately scaled and centred.

## Keywords

Traveling salesman problem well-connected cities location dependent edge lengths## 2000 Mathematics Subject Classification

Primary: 60J10 60K35 Secondary: 60C05 62E10 90B15 91D30## Notes

### Acknowledgements

The author would like to thank Professors Rahul Roy, Thomas Mountford, Federico Camia and the referee for crucial comments that led to an improvement of the paper. He would also like to thank Professors Rahul Roy, Thomas Mountford and Federio Camia for his fellowships.

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