I just started learning about linear programming in my class, and I'm having some trouble understanding the MST Formulation Integer Linear Programming (Cutset Formulation).

This is the definition:

> 
> An *integer* linear program that solves the minimum spanning tree problem is as follows:
> 
> 
> $$
> \begin{align\*}
> \text{Minimize} & \sum\_{\{u,v\} \in E} w\_{u,v}x\_{u,v} \\
> \text{subject to} & \sum\_{\substack{\{u,v\} \in E\colon \\ u \in S, v \notin S}} x\_{u,v} \geq 1 & \text{for all $S \subseteq V$ with $0 < |S| < |V|$} \\
> & \sum\_{\{u,v\} \in E} x\_{u,v} \leq |V|-1 \\
> & x\_{u,v} \in \{0,1\} & \text{for all $\{u,v\} \in E$}
> \end{align\*}
> $$
> 
> 
> 

$x\_{ij}$ is defined to be $1$ if the edge $ij$ is in $T$.

What exactly does the optimum value represent here? Is it the number of edges that are used in our MST? If so, how does knowing the number of edges help us in forming the MST?

How does this ILP "work" to solve the MST problem?