The bright side of simple heuristics for the TSP

TL;DR

This paper shows that simple heuristics like greedy and nearest-neighbor for the Traveling Salesman Problem (TSP) perform much better on average than worst-case bounds suggest, especially for random points in regular metric spaces.

Contribution

It demonstrates that the worst-case approximation factors for these heuristics are only achieved when the optimal tour is very short, and provides bounds on their additive errors for random points in regular metric spaces.

Findings

Heuristics have small additive errors relative to the optimal tour length for random points.

Worst-case approximation factors are only realized when the optimal tour is unusually short.

Results apply to points in fixed d-Ahlfors regular metric spaces, including fractals and manifolds.

Abstract

The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal on the order of the \emph{optimal} tour length through \emph{random} points in the same space, for $d>1$.