The dual has a solution $(w^*,\pi^*)$ and $\sum_{l}w^*_l+\sum_f\pi^*_f=\sum_{l,f}v_{lf}x^*_{lf}$.

The following observations are consequences of complementary slackness.

• If $x^*_{lf}=1$ then $w^*_l+\pi^*_f=v_{lf}$.
• If $l\in\ll$ is unmatched, then $w_l=0$.
• If $f\in\ff$ is unmatched, then $\pi_f=0$.

Theorem. $(x^*,w^*,\pi^*)$ is a stable allocation iff $x^*$ is an optimal matching and $(w^*,\pi^*)$ solves the dual lp.