Suppose given $n$ pair of points $P=\{(p_1,q_1),\dots,(p_n,q_n)\}$ in the plane that each pair $(p_i,q_i)\in \mathbb{R}^2$ can't belong to the same group. We want to partition points into $K$ groups such that we minimize function $f:x\rightarrow \mathbb{R}$. So we try to find an efficient algorithm. We want a running time of $O(n^3)$ or better.
My attempt: I find a black box that can partition $2n$ points in the plane in $O(n\log n)$ and minimize $f(n)$, without any constraint. But at this step a get stuck that how I can use black box?
