In optimal transport, we calculate the distance between two probability measures $\mu$ and $\nu$ over the compact set $[a,b]\subset\mathbb R$, using the Earth Movers distance which is a special case of the Wasserstein formula:
$$W=\inf_\pi\int|x-y|\,{\rm d}\pi(x,y),$$
where $\pi$ is interpreted as a transport matrix satisfying $\int\pi(x,y)dy=\mu(x)$ and $\int\pi(y,x)dy=\nu(x)$.
It is said that the Earth Movers distance shown above is solved using the Hungarian algorithm by Munkres, which is also used for solving the Assignment Problem. Are optimal transport and the assignment problem related then, and how mathematically?
