Before 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 . In this paper, we propose the first polynomial-time approximation scheme for Max TSP in  arbitrary metric space of fixed doubling dimension.
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 a vector norm in fixed-dimensional real space. We propose the first polynomial-time approximation scheme for Max TSP in arbitrary metric space of fixed doubling dimension.