We are given a bipartite graph of $n \leq 200$ vertices in both the first and the second partite set. Let $U$ be some set of vertices in the first set, and $V$ those vertices from the second that are conected to some of the vertices from $U$. If for every choice of $U$ we have $|U| \leq |V|$ (Hall's condition) then there exists a perfect matching (Hall's theorem).
But in our graph we know there is no such matching. That means there exists some set of vertices $U$ violating Hall's condition, and I'd like to find it with complexity around $O(n^3)$ - hints instead of full solutions are most welcome.
What I already tried was finding the maximum matching first, and then searching for our subset, but I couldn't figure out how to do this. Also, I thought of ways of reducing this problem to some max-flow (as you do with the maximum matching) but it also seemed to me like a dead end.