The heaviest induced subgraph problem I am interested in such a combinatorial problem: given a graph $G=(V, E)$ and a weight functions $w_v: V \mapsto R$, and $w_e: E \mapsto R$ we are asking about such a induced subgraph $G' = (V', E')$ of $G$ that maximizes the sum:
$
\sum_{e \in E'} w_e(e) + \sum_{v \in V'} w_v(v)
$.

The problem is NP-Hard (by the reduction from maximum clique problem) so any suggestions for approximation solutions (even greedy) and links to the literature would be appreciated.
$\newcommand{\R}{\mathbb{R}}$
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.