{cite}`boerma2023composite` propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.

Their approach relies on *Complementary Slackness*: given a primal solution $\mu$, $(\phi , \psi) $ is a dual solution if and only if for all $x \in X$ and $y \in Y$

* $\phi_x + \psi_y \leq c_{xy}$ (dual feasibility),

* $\phi_x + \psi_y = c_{xy}$ if $\mu_{xy}>0$ (complementary slackness).

Let's generate an instance and compute the optimal matching.

```{code-cell} ipython3