I have a complete bipartite graph. Each edge of the graph has a length $l$ and weight $w$.
My goal is to find a complete subgraph (a subgraph that is a complete bipartite graph itself) of this bipartite graph such that the weighted average length ($\sum wl/\sum w$) of the edges in the subgraph is maximized.
(Edit: an additional constraint of the subgraph is that there is one given edge, say edge $ij$, and its two vertices $i$ and $j$, that must be included in the subgraph.)
Any ideas on how to solve this problem? Or is it a NP-hard problem? Thanks.