The 2-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces two edges of the tour by two other edges, as long as this yields a shorter tour. We will prove that for Euclidean Traveling Salesman Problems with $n$ cities the approximation ratio of the 2-Opt heuristic is $\Theta(\log n/ \log \log n)$. This improves the upper bound of $O(\log n$) given by Chandra, Karloff, and Tovey [3] in 1999.

翻译：2-Opt Heuristic是旅行推销员问题的一个简单的改进图。 它首先是一个任意的旅游, 然后反复用另外两个边缘取代旅游的两边, 只要这能带来更短的旅游。 我们将证明对于欧洲旅游推销员问题, 2-Opt Heuristic 和 $n 城市的近似比率是 $\ Theta (\log n/\log\log\log n) 。 这改善了Chandra、 Karloff 和 Tovey 在1999年给出的 $O (\log n$) 的上限 。