Maximum weighted subgraph Problem: Given a graph G with positive integral weights on nodes as well as edges, find an induced subgraph H with maximum total weight per node, i.e., the goal is to maximize
$$
\frac{ \sum_{e \in H}w(e) + \sum_{v \in H} w(v) }{|V(H)|}.$$
If you allow negative weights, then it is NP-hard (easy reduction from Clique). I have a strong feeling that the above is not NP-Hard.
I would love to have some first-thoughts on the above, if not a full-fledged optimal algorithm. Does it seem to be NP-Hard? What is the closest known problem -- my thoughts were Prize Collection, Clique -- but both seem much harder.
The Dense Subgraph Problem (thanks @PalGD) is a special case of the above problem--with no node weights and unit edge weights. See the solutions for the dense subgraph problem here. These solutions (LP and 2-approximation Greedy) easily generalize to the above problem (generalizing LP requires a bit of understanding/effort, but generalizing Greedy is trivial).