The (1,2)-TSP is a special case of the TSP where each edge has cost either 1 or 2. In this paper we give a lower bound of $\frac{3}{2}$ for the approximation ratio of the 2-Opt algorithm for the (1,2)-TSP. Moreover, we show that the 3-Opt algorithm has an exact approximation ratio of $\frac{11}{8}$ for the (1,2)-TSP. Furthermore, we introduce the 3-Opt++-algorithm, an improved version of the 3-Opt algorithm for the (1-2)-TSP with an exact approximation ratio of $\frac{4}{3}$.

中文翻译：

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关于（1,2）-TSP的3-Opt算法的逼近率

（1,2）-TSP是TSP的特例，其中每个边的成本为1或2。 （1,2）-TSP的2-Opt算法。此外，我们证明了3-Opt算法对于（1,2）-TSP具有精确的近似值\\ frac {11} {8} $。此外，我们介绍了3-Opt ++-算法，它是（1-2）-TSP的3-Opt算法的改进版本，精确近似比率为\\ frac {4} {3} $。