# Find a minimum-cardinality Hall-violator

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?