Is there a known algorithm (besides brute force) for the following problem:
We have given an edge-weighted complete graph $G$ and a finite set of natural numbers $A = \lbrace n_1,\ldots,n_k \rbrace$ (to begin with we can just assume that $A$ has only one element). We want to decompose $G$ in cliques whose size is an element of $A$ and such that the sum of all edges of these cliques is maximal. (If this is not possible, e.g., in the case when $A = \lbrace n \rbrace$ and $n$ is not a divisor of the number of vertices of $G$, we are allowed to add vertices and edges (with weight $0$) to $G$.)
This is a modeling for the following problem: Given a group of people, a "friendship index" between each two people and a hotel with rooms of size $n_1,\ldots,n_k$, determine the optimal room division.
I have posted this question (some time ago) also at MathStackExchange, but without answer.