Let $G = (V, E, W)$ be a weighted graph with positive and negative weights. I would like to find the set of vertices $V^\prime$ such that the sum of the weights of the edges that they share is the maximum you can get from $G$. I suspect that this is an NP-complete problem, so my question is whether this can be reduced to a known NP-complete problem. Since $V^\prime$ does not necessarily have to be the minimum subset that satisfies the constraint, I think I can't use the solutions for the vertex/set cover problems, so any ideas would be welcome.
An easier problem to use would be max clique. We do the transformation as follows. We take the graph that we wish to find the max clique in and give all edges in it weight 1, and for all non edges we give weight (-)infinity (or if you dont like infinity an arbitrarily large number based on the number of edges total, negative enough so that if one is picked the total will always be negative). This problems solution will give us a clique since the negative infinity weight will prevent selecting anything that does not have edges between them for the solution. This clique is also the largest, as a larger clique would have a larger edge sum and we assume this algorithm returned the largest edge sum. This is therefor at least as hard as solving max clique, which is NP complete.