Diffusive behavior of a greedy traveling salesman

TL;DR

This study uses Monte Carlo simulations to analyze the diffusive behavior of a greedy algorithm in the traveling salesman problem across different dimensions, revealing dimension-dependent diffusive properties and potential links to animal foraging strategies.

Contribution

It demonstrates how the diffusive properties of a greedy TSP algorithm vary with dimension, highlighting a possible critical dimension at d=2 and connecting to Levy flight behaviors.

Findings

For d=3 and 4, <r^2> scales linearly with steps t.

In d=2, scaling includes logarithmic corrections.

Distribution of lengths differs significantly between d=2 and higher dimensions.

Abstract

Using Monte Carlo simulations we examine the diffusive properties of the greedy algorithm in the d-dimensional traveling salesman problem. Our results show that for d=3 and 4 the average squared distance from the origin <r^2> is proportional to the number of steps t. In the d=2 case such a scaling is modified with some logarithmic corrections, which might suggest that d=2 is the critical dimension of the problem. The distribution of lengths also shows marked differences between d=2 and d>2 versions. A simple strategy adopted by the salesman might resemble strategies chosen by some foraging and hunting animals, for which anomalous diffusive behavior has recently been reported and interpreted in terms of Levy flights. Our results suggest that broad and Levy-like distributions in such systems might appear due to dimension-dependent properties of a search space.