  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 $\frac{4}{3}$.
