Let $G = (V,E)$ be an undirected graph, with $e \in E$ has positive weight $w_e$.
Given a set of integers $I = \{i_1,\dots,i_n\}$ such that $\sum_{k=1}^n i_k = |V|$. I want to find a partition $P$ of $G$ of cardinality $n$ such that each element of the partition has cardinality corresponding exactly to one of the elements of $I$.
The maximization function sums all the edges weights $w_{(v_1,v_2)}$ if both $v_1$ and $v_2$ belong the same subset within the solution - i.e. $v_1, v_2 \in P_j$, for any $P_j \in P$.
Similarly, the objective function might be minimizing all the weights $w_{(v_1,v_2)}$, where $v_1$ and $v_2$ belong different element of the partition.
My question is whether such a problem has already an official formulation in literature. Or if there is an interesting equivalent problem.