Before revision:  This problem is APX-hard in the general and metric cases but admits efficient approximation schemes in the geometric setting, when the vertices of the input graph are some points in fixed-dimensional real space and the edge weights are induced by some vector norm.
After revision:  This problem is APX-hard in the general and metric cases but admits approximation schemes in the geometric setting, when the vertices of the input graph are some points in fixed-dimensional real space and the edge weights are induced by some vector norm.