Formalizing and optimizing constraints involving booleans, pairs of booleans, and integer sums
Can this be reformulated as follows (careful, I am sleep-deprived with a cranky toddler in the room):

We are looking at a random graph $G$ with 6000 weighted nodes, a subgraph $G'$ induced by about 4000 edges, and 140.000 other weighted edges of positive weight but strictly smaller than the nodes they connect to.

A satisfying assignment for $C$ induces a bipartite subgraph of $G'$ (all nodes with assignment 1 plus their immediate neighbors with assignment 0). So you are looking to optimize over the set of bipartite subgraphs of $G'$?

A few random thoughts:


How structured are $G$ and $G'$? I presume a random graph model is not a very good model? What is the distribution of degrees of nodes?
Are all nodes in $G'$?


Is there a way we can bound the sum of all $w_2$-induced weights for given node? We know that the node weight $w_1(v)$ is bigger than the edge weight $w_2(v, \centerdot)$, is there something to be said for $\sum_{v_2 \in V} w_2(v, v_2)$? That could give some information about whether "switching a node on" is worth it...

If nodes have a tendency to outweigh the sum of their edges, would that hint at a bigger bipartite subgraph being always better than a smaller one, and at the problem becoming one of maximizing the size of the bipartite subgraph?