Find permutation minimizing distance
Your problem appears to be very close to the minimum quadratic assignment problem, see for example a paper by Hassin, Levin and Sviridenko; your problem is close to the metric variant (since $D$ is a metric). When $D_{i,j} = |i-j|$ we get the NP-hard problem minimum linear assignment, whose best known approximation ratio is $\tilde{O}(\sqrt{\log n})$.