Has this problem been studied?
We start with a complete, simple, undirected graph with edge weights. The problem is to delete edges so as to partition the graph into complete disjoint subgraphs while maximizing the sum of the edge weights over the whole graph (deleted edges not being counted).
There is no constraint on the size or number of subgraphs in the solution.
Note this is interesting only if there are negative edge weights. Otherwise the solution would be to take the whole graph.
Clearly we would like to delete all the negative-weight edges while keeping all positive-weight edges, but the requirement that disjoint subgraphs be complete makes a trade-off between these goals.
I have seen discussion of similar graph partitioning problems but not this exact one. (Eg it is a bit like the maximum clique problem, but here I want to put every node in a disjoint clique, in the optimal way). I am guessing it is NP-complete. I am interested to know where to start looking for efficient heuristics.