Improved Smoothed Analysis of 2-Opt for the Euclidean TSP

TL;DR

This paper provides a comprehensive smoothed analysis of the 2-opt heuristic for the Euclidean TSP, establishing polynomial bounds for all dimensions and improving existing complexity results.

Contribution

It offers the first direct Gaussian smoothed complexity bounds for all dimensions in Euclidean TSP, enhancing previous results and unifying the analysis across dimensions.

Findings

Polynomial smoothed complexity bounds for all dimensions d

Improved bounds over previous analyses for Euclidean 2-opt

Unified analysis applicable to Gaussian perturbations in all dimensions

Abstract

The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case running time is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey \& Veenstra, who obtained smoothed complexity bounds polynomial in $n$, the dimension $d$, and the perturbation strength $\sigma^{-1}$. However, their analysis only works for $d \geq 4$. The only previous analysis for $d \leq 3$ was performed by Englert, R\"oglin \& V\"ocking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in $n$ and $\sigma^{-d}$, and super-exponential in $d$. As no direct analysis existed for Gaussian perturbations that yields polynomial bounds for all $d$, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt.