Traveling salesman problem across dense cities

TL;DR

This paper analyzes the traveling salesman problem (TSP) in dense city models, providing variance estimates and convergence results for the TSP length under specific connectivity and density conditions, and extends findings to the unconstrained case.

Contribution

It introduces new variance and convergence estimates for TSP lengths in dense city models and extends these results to the unconstrained node distribution case.

Findings

Variance estimates for TSP length in dense city models

Convergence of scaled TSP length to zero in probability under certain conditions

Large deviation estimates for TSP length

Abstract

Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and let~\(TSPC_n\) denote the length of the minimum length cycle containing all the~\(n\) nodes, corresponding to the traveling salesman problem (TSP). We obtain variance estimates for~\(TSPC_n\) and prove that if the cities are well-connected and densely populated in a certain sense, then~\(TSPC_n\) appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for~\(TSPC_n.\) Using the proof techniques, we alternately obtain corresponding results for the length~\(TSP_n\) of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square~\(S.\)