Let's say you are given a set with $V$ vectors and you are given a simple Euclidean distance metric $d(a,b)$ for $a, b \in V$. Then what would be the most efficient algorithm to build a subset $S$ of a desired size $k$ such that the sum of the distances between the elements of the subset is maximized?
In other words, how does one find a subset with the most distinct vectors?
I have tried modeling this as a complete graph where the nodes are vectors and the edges the Euclidean distances between them, but I couldn't find a graph algorithm that solves this problem. It seems like finding an independent set of given size $|S|$ with the maximum number distance between them.