A real matrix is called totally positive if all of its minors are positive. Of course, to check that a given matrix is totally positive it is not necessary to check all the minors: for example, it suffices to check that all initial minors are positive. A minor $A\begin{pmatrix}I\\J\end{pmatrix}$ of $A$, corresponding to the sets of row indices $I$ and column indices $J$ is called initial if both $I$ and $J$ consist of consecutive indices and $1\in I\cup J$ (so every entry of $A$ is the lower right entry of a unique initial minor). In fact, there are many distinct families of minors such that their positivity implies total positivity, parametrized by "double wiring diagrams" (see https://link.springer.com/article/10.1007/BF03024444).
There are similar tests for total non-negativity. But all such tests seem to optimize the number of minors to be checked and not the total computational cost. One would imagine that checking more smaller minors could in some cases be done faster then checking fewer larger minors. Has any work been done in this direction?
