# #StackBounty: #np-hard #matching #bipartite-matching Find a minimum-cardinality Hall-violator

### Bounty: 50

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?

Get this bounty!!!

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