Let $G$ be a graph of $n$ arcs and let $x\in \mathbb{R}^n$. I want to compute the orthogonal projection of $x$ onto the set of radial graphs with $k$ roots contained in $G$ (or a forest with $k$ root) i.e. $$\Pi \left( x\right) =\underset{y\in S}{\arg \min }\left\Vert x-y\right\Vert ^{2},$$ where $S$ is the underlying set.
My question is: how is this projection linked to the minimal spanning trees? I really appreciate any help you can provide.
I think you mean that $y$ is the characteristic vector. That is, $y_{ij}=1$ if edge $(i,j)$ is in the forest and $0$ otherwise. Let $E$ be the edge set of $G$. Given $x$, you want to find $y$ to minimize $$ \sum_{(i,j)\in E} (x_{ij}-y_{ij})^2 = \sum_{(i,j)\in E} (x_{ij}^2-2x_{ij}y_{ij}+y_{ij}^2) = \sum_{(i,j)\in E} x_{ij}^2 + \sum_{(i,j)\in E} (1-2x_{ij})y_{ij}, $$ which is equivalent to minimizing $$\sum_{(i,j)\in E} (1-2x_{ij})y_{ij},$$ which is the objective function for the problem of finding a minimum forest with edge weights $1-2x_{ij}$.
