Let $T=(V,E)$ denote a tree. Each node $j \in V$ in the tree has a known attribute $c_j$.
From T, construct a bi-directional graph $G' = (V, E')$ where $E' = \{(j,k), (k,j)| (j,k) \in E\}$. Simply, make each edge bi-directional.
Let us define a binary decision variable $x_{jk} =1$ for $(j,k) \in E'$ if we keep an arc from node $j$ to node $k$ and zero otherwise.
Consider the following optimization problem
\begin{equation} \begin{array}{rl} F(x):= \min_{x\in\mathbb{R}^n}\ & \sum_{k \in V} (c_k - \sum_{(j,k) \in E'} c_j x_{jk})^2\\ \text{s.t.}\ & x_{jk} x_{kj} =0 \quad \forall (j,k) \in E'. \end{array} \end{equation}
Intuitively, the model aims to determine the best direction and an associated weight for each edge such that the objective function is minimized.
Can we perhaps show that this problem is NP-hard? Any NP-hard problem that can be reduced to this optimization problem?