Given: A matrix $A \in \mathbb R^{n \times n}$
Wanted: A topological sort $\pi$ of $\{1,\ldots, n\}$ that satisfies
$$ \text{arg}\min_{\pi \in \Pi}\max\limits_{\substack{(i,j) \in V \times V: \\ \pi(i)> \pi(j)}}a_{ij}=\max\limits_{\substack{(i,j) \in V \times V: \\ \pi(i)> \pi(j)}}a_{ij}. $$
So let me give an example: Imagine we have given the topological sort $\pi=\{1,\ldots,d\}$, then we have
$$ \max\limits_{\substack{(i,j) \in V \times V: \\ \pi(i)> \pi(j)}}a_{ij}=\max\limits_{\substack{(i,j) \in V \times V: \\ i> j}}a_{ij}, $$
so the value for $\pi$ would be just given by the maximum value in the lower triagle of the matrix $A$. We want to find a $\pi$ such that this maximum value is minimized; Now does this problem already exist?
This problem is also connected to graph theory; Imagine that $\{1,\ldots, n\}$ denote some nodes; Then if we assume $A$ to be the matrix of edge weights, then the minimization problem would be to find a topological sort such that the maximum edge weight which is in the opposite way of the topolocal sort is minimized...
I know that it can be solved fairly easy but I am curious if the problem is already known..