The shortest Hamiltonian path (solution) for a set of points in $\mathbb{R}^k$ (in Euclidean space) changes subject to $k$.
For example if for $k=1$, the shortest Hamiltonian path will be the sorted set of points, which can be achieved in $O(n\log n)$.
My question is what happens when $k$ increases - does the solution become more and more complex or is it the same for all values of $k$ larger than $1$?