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Find a minimum-cardinality Hall-violator
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Problem
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?
$S \subseteq X$ 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?
Solution
This problem is $\mathsf{NP}$-hard. The problem is also known as Hall set problem and there is a reduction from Clique problem. See Theorem 3.2.5 from this thesis.
I was thinking along the direction of the Minimum $k$-Union problem. Then, I came across this thesis.
I was thinking along the direction of the Minimum $k$-Union problem. Then, I came across this thesis.
Context
StackExchange Computer Science Q#80210, answer score: 3
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