Just a thought I had in mind. I can use classical matrix multiplication to compute min-plus matrix multiplication. Generally speaking, considering $(n+1)^{a_{i,j}}$ for each entry and then taking the smallest value.
But what about the other way around? I want to compute classic matrix multiplication by using as a blackbox an algorithm that computes min-plus. Not really sure which ideas... (And what would be the runtime)
Basically, you want to compute $\hat{c}_{i,j} = \min\limits_{k=1}^n \{a_{i,j} + b_{i,j}\}$ from a blackbox (classical matrix multiplication), which computes $c_{i,j} = \sum\limits_{k=1}^n a_{i,j} \times b_{i,j}$. So basically, we need to convert a multiplication into an addition operation and a minimum into a summation. You need to process the input elements so that the desired effect can be achieved.
Converting multiplication into an addition:
exploit the $\log$ operation
Converting the minimum into a summation:
exploit the $L_\infty$ norm
