The heaviest induced subgraph problem
$\newcommand{\R}{\mathbb{R}}$
Let's consider a special case of the problem where vertex has negative weight and edges have positive weight. 
So $w_v: V\to \R^-$ and $w_e: E\to \R^+$.

Finding the heaviest induced subgraph is equivalent to a min-$st$-cut computation on a suitable graph. We will refer to the slides about densest subgraph in this presentation. 

Indeed, minimizing $w_e(E(V'))+w_v(V')$ is equivalent to minimizing $(-w_v(V')) + \frac{1}{2} w_e(E(V',\bar{V'})) + \frac{1}{2} \sum_{v\in \bar{V'}} \deg(v)$. (Note the degree here are weighted degree, that is $\deg(v) =\sum_{e:v\in e} w_e(e)$) 
The derivation of the above fact is similar to slide 21.
Then, this can be solved easily by modeling it as a min-$st$-cut in some other graph (see slide 22). It is crucial to have negative vertex weights and positive edge weights for the reduction to work.