center selection problem: Simple greedy algorithm
Given a set of points, the task is to find a set of centers minimizing the covering radius, defined below. The centers themselves could be arbitrary points in space, or sometimes must be selected among the points or among a set of potential centers.

The covering radius is the minimum $r$ such that each point in the set is at distance at most $r$ from a center. Clearly, adding centers can only reduce the radius, and in general it does. As a very extreme examples, given $n$ distinct points, for any $n-1$ centers that we choose, the covering radius will be positive, while if we take all the points as centers, the covering radius drops to zero.

As another example, consider three points on the real line at locations $-1,0,1$. The best choice for a single center is $0$, with covering radius $1$. The best choice for two centers is $-1/2,1/2$, with covering radius $1/2$. However, if we first put a center at $0$, no choice for the second center will reduce the covering radius.