I have been recently researching Happynet in terms of approximation and I have found out that there is a little interest in this topic.
What's the reason for this? Are there any related problems, that get better attention?
The Happynet problem is related to minimizing the energy in a Hopfield net.
The problem is defined as follows: Given a graph $G = (V,E)$ and edge weights $w: E \to \mathbb{Z}$, find a function $s: V \to \{-1,1\}$ such that for all $v \in V$, the sum $$\sum_{u: \{v,u\} \in E} s(v)s(u)w(\{u,v\})$$ is non-negative.
Put otherwise, the task is to find a cut in an edge-weighted graph $G$ such that for each node $v$ the total weight of cut-edges incident to $v$ is at most as large as the total weight of non-cut-edges incident to $v$.
Curiously, a feasible solution always exists. However, it seems that no polynomial-time algorithm is known for solving the problem. For more information, see e.g. Giannakos and Pottie (2005).