I would like to know if the following problem has already been studied, and if so how is it called. In particular I'm interested in approximability results.
Input: A complete graph G with non-negative integer weights on edges and an integer $K\ge 2$.
Output: A $K$-partition $P=\{P_1, ..., P_K\}$ of $V(G)$
Measure (to maximize): The sum of the weights for the edges with both endpoints in the same set of $P$, i.e.:
$$
M(P)=\sum_{i=1}^k \sum_{u,v\in P_i} w\left( u,v \right)
$$
where $w$ is the edge-weight function.
Thanks in advance for any suggestion.