Does the TSP (traveling salesman problem) change in difficulty when the number of dimensions is greater than the number of cities?

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?

• Graph theory does have a notion of dimensionality to it, in that a graph may or may not be embeddable into $\mathbb{R}^n$ for certain $n$ (where "embedabble" means it can be drawn without intersecting edges). The smallest $n$ for which this can be done can be interpreted as the dimension of the graph itself. This dimensionality enters into certain problems (for example, coloring problems). It doesn't really enter into the TSP in any significant way, however.
• What does affect TSP is whether the edge costs are metric: whether they satisfy the triangle inequality $d(u,w) \le d(u,v) + d(v,w)$. If the graph is embeddable in $\mathbb R^n$ for any $n$, then the edge costs (Euclidean distances) are metric, which allows for some approximation algorithms that do not work in general. Dec 1, 2018 at 22:29
• Except of course for $d=1$: If the cities are points of $\Bbb R$ with the euclidean distance then the TSP is pretty easy... Dec 2, 2018 at 18:49