Given a set of points in $\mathbb{R}^2$, partition them into $2$ clusters such that the maximum diameter of all the clusters gets minimized. The definition of cluster diameter is given below:
- The diameter of a cluster is the diameter of the minimum enclosing circle of the points on that cluster. ($2$-center clustering)
- The diameter of a cluster is the maximum possible distance between any two points on that cluster. (max diameter $2$-clustering)
Consider, we have an exact solution for max diameter $2$-clustering. We know that these two clusterings are not equivalent. However, if we solve max diameter $2$-clustering, how much it approximates the $2$-center clustering? Is there any way we can find the exact $2$-center clustering from the exact solution of max diameter $2$-clustering?