I'm a working software engineer faced with the following problem: I have a set of points on a 2d plane. Each point can have one of $k$ different colours. I wish to select one point of each colour that are "as close as possible". To formalize that a bit, let's say that they are all covered by a circle with the smallest possible radius.
I have one idea for a solution: Fix some point $p$ in the set of points. For each colour $c$, choose a point of colour $c$ that is closest to $p$. Do this (from scratch) for every point p, and eventually take the solution with minimum radius.
The question I'm faced with now is: How good a solution is this? Does it find the exact answer? Is it an approximation? If so, what is it's approximation ratio?
Edit: So this has a counterexample. The question is then is this a good approximation?