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 $\frac{4}{3}$.

翻译：(1,2)-TSP是TSP的一个特例,其中,每一边缘的成本为1或2。 在本文件中,我们对2-2-2-TSP算法的近似比率(1,2)-TSP,我们给出了一个较低的约束值,即$\frac{3 ⁇ 2美元。此外,我们表明,3-opt 算法对(1,2)-TSP的精确近似比率为$\frac{11 ⁇ 8美元。此外,我们引入了3-Opt++-algorithm,这是1-2)-TSP的3-Opt算法的改进版,其精确近似比率为$\frac{4}3美元。