Given a bipartite graph $ (X,Y，E)$ , in which there is no perfect matching, I want to find a smallest subset that violates Hall’s condition, i.e., a minimum-cardinality set $ S \subseteq X$ for which $ |N(S)|<|S|$ .

This problem is the optimization version of a former question Finding a subset in bipartite graph violating Hall's condition, from which I know there exists a polynomial-time algorithm for finding such $ S \subseteq X$ . Does there exists a polynomial algorithm for the optimization problem?