The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the well-known Lin-Kernighan algorithm and often used in practice. So far, only upper and lower bounds on the approximation ratio of the 2-Opt heuristic for the metric TSP were known. We prove that for the metric TSP with $n$ cities, the approximation ratio of the 2-Opt heuristic is $\sqrt{n/2}$ and that this bound is tight.

中文翻译：

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度量旅行商问题的2-Opt启发式逼近比

2-Opt 启发式算法是寻找度量旅行商问题的最佳解决方案的最简单算法之一。它是著名的 Lin-Kernighan 算法的关键成分，经常在实践中使用。到目前为止，仅知道度量 TSP 的 2-Opt 启发式的近似比的上限和下限。我们证明对于具有 $n$ 个城市的度量 TSP，2-Opt 启发式的近似比率是 $\sqrt{n/2}$ 并且这个界限很紧。