Euclidean traveling salesman problem with location dependent and power weighted edges

TL;DR

This paper studies a generalized Euclidean TSP with location-dependent and non-standard edge weights, proving convergence of the minimal cycle weight and providing deviation bounds as the number of nodes grows large.

Contribution

It introduces a framework for analyzing Euclidean TSP with complex, location-dependent weights and establishes convergence and bounds under certain conditions.

Findings

TSP minimal weight scaled and centered converges to zero almost surely and in mean.

Provides upper and lower deviation bounds for the TSP weight.

Extends classical Euclidean TSP results to more general weight functions.

Abstract

Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f\) and let~\(K_n\) be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge~\(e\) in~\(K_n\) we associate a weight~\(w(e)\) that may depend on the \emph{individual locations} of the endvertices of~\(e\) and is not necessarily a power of the Euclidean length of~\(e.\) Denoting~\(TSP_n\) to be the minimum weight of a spanning cycle of~\(K_n\) corresponding to the travelling salesman problem (TSP) and assuming an equivalence condition on the weight function~\(w(.),\) we prove that~\(TSP_n\) appropriately scaled and centred converges to zero a.s.\ and in mean as~\(n \rightarrow \infty.\) We also obtain upper and lower bound deviation estimates for~\(TSP_n.\)