Consider the LP
\begin{equation} \begin{aligned} v_P(\ll\cup\ff)&=\max_x\sum_{l,f}v_{lf}x_{lf}\\ \text{s.t.}\qquad&\begin{aligned}[t] \sum_fx_{lf}&\leq 1\text{ for all $l\in\ll$,}\\ \sum_lx_{lf}&\leq 1\text{ for all $f\in\ff$,}\\ x_{lf}&\geq 0\text{ for all $l\in\ll,f\in\ff$.} \end{aligned} \end{aligned} \end{equation}