The set $C$ of all vectors satisfying the constraints is a convex polytope, the ``fractional matchings''.

Theorem. (Birkhoff-von Neuman). $x$ is a vertex of $C$ iff for all $l$ and $f$, $x_{lf}\in\{0,1\}$.

Corollary. $x^*$ is an optimal matching iff it is a basic optimal solution to the LP.