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Finding the minimum sum of a subset of entries of a given matrix with combinatorial constraints

Given a matrix $M\in\mathbb{N}^{n\times n}$, let $Z$ be the set of all the $M$'s entry subsets $S$ such that (i) no two entries of $S$ are on the same row or column of $M$ and (ii) $|S|=n$. Clearly we ...