Approximation Ratio of Local search for $k-$center problem
Local search (with a single swap) doesn't give you a good approximation factor in the worst case for $k$-center, as illustrated by the following example.

Take a simplex in $\mathbb{R}^{k-1}$, and put $k$ points at each of the $k$ vertices, for a total of $k^2$ points.

Note that between each pair of vertices of the simplex the distance is 1.  The optimal solution takes one point from each vertex, obtaining cost zero.  And that's the only way to obtain cost zero.. any other type of solution has cost 1.

So if you start local search with $C$ being any solution that is more than one swap away from such an optimal solution, no local improvement (in one step) is possible.  For example, start local search with $C$ containing all $k$ of the points at a single vertex.  That solution has cost 1, and no single swap (or even $k-2$ swaps) can reduce the cost below 1.