On the approximation ratio of the 3-Opt algorithm for the (1,2)-TSP

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 lower bound 32 for approximation ratio 2-Opt algorithm (1,2)-TSP. Moreover, show that 3-Opt an exact 118 Furthermore, introduce 3-Opt++-algorithm, improved version (1-2)-TSP with 43.