#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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