Travelling Salesman Problem with a Center

TL;DR

This paper investigates a modified travelling salesman problem that incorporates a centrality measure into its cost function, revealing phase transitions affecting path shape, cost scaling, and optimization efficiency.

Contribution

It introduces a new TSP variant with a centrality component and analyzes its phase transition behavior using simulated annealing.

Findings

Identifies a phase transition separating two distinct path regimes.

Shows the transition affects the scaling of path length and centrality measure.

Demonstrates the impact on simulated annealing efficiency and path shape.

Abstract

We study a travelling salesman problem where the path is optimized with a cost function that includes its length $L$ as well as a certain measure $C$ of its distance from the geometrical center of the graph. Using simulated annealing (SA) we show that such a problem has a transition point that separates two phases differing in the scaling behaviour of $L$ and $C$, in efficiency of SA, and in the shape of minimal paths.