Differential Approximation Results for the Traveling Salesman Problem with Distances 1 and 2

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any 2 > 0, it is NP-hard to approximate both problems better than within 741/742 + 2. The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.