Optimal assignments with supervisions

In this talk I provide a graph-theoretic proof of the tropical Jacobi identity, alternative 

to the matrix-theoretic proof recently obtained jointly with Akian and Gaubert. The latter was inspired by the classical identity: 

The (J^c,I^c)-minor of a matrix A corresponds, in some way to be defined, to the (I,J)-minor of A^{-1}). 

That is, the compound of order k of A corresponds to the compound of order (n-k) of its inverse.

We provide an application of this theorem to optimal assignments with supervisions.
More precisely, we consider the problem of assigning multiple tasks to one team, or daily tasks to 
multiple teams, where each team has a supervisor-task or a supervised task. 
This is a joint work with Marie Maccaig and Sergey Sergeev.