The Approximation Ratio of the $k$-Opt Heuristic for the Euclidean Traveling Salesman Problem

Abstract

The $k$-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces $k$ edges of the tour by $k$ other edges, as long as this yields a shorter tour. We will prove that for 2-dimensional Euclidean Traveling Salesman Problems with $n$ cities the approximation ratio of the $k$-Opt heuristic is $\Theta (\log n/\log \log n)$. This improves the upper bound of $O(\log n)$ given by Chandra, Karloff, and Tovey in 1999 and provides for the first time a non-trivial lower bound for the case $k\ge 3$. Our results not only hold for the Euclidean norm but extend to arbitrary $p$-norms with $1\le p<\infty$.

this article supersedes arXiv:2010.02583

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