The title pretty much says it all.
I am curious if the TSP is dependent only upon the number of cities, $c$, involved, or if the dimensionality, $d$, matters to the problem.
For example, if $d>>c$ does the problem shift in some manner, making the time to solve the problem less dependent on the number of cities?
The traveling salesman problem is, in its basic form, a graph theory problem. All vertices (the cities to visit) are connected, and all edges are assigned a "distance", and you need to answer: "what is the shortest route that starts at a given vertex, visits all other vertices, and comes back to the start point?"
The only thing that changes in a multi-dimensional format is that calculating these distances takes more variables, but this is calculated in linear time (I'm assuming Euclidean distances). But once you get calculate them all and get the graph described above, the problem is exactly the same. So no change in complexity.