Consider the set cover problem: given a collection of sets ${\cal U}$ whose elements come from $\{1, \ldots, m\}$ find the smallest number of sets in ${\cal U}$ whose union is all of $\{1, \ldots, m\}$. My question is: what would be a certificate for this problem? That is, if I wanted to convince someone that it is not possible to do this with fewer than $k$ sets, what information could I provide that would allow someone to verify that this is the case in polynomial time?
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The set cover is not a decision problem, so it doesn't necessarily have a certificate. Rather, it is an optimization problem.
To make it a problem in NP (a decision problem, with a certificate), you need to apply the standard trick to convert an optimization problem to a decision problem: introduce a budget. Thus, the decision budget is: given a collection of sets and an integer $k$, determine whether there exists a subset of $\le k$ sets whose union is all of $\{1,\dots,m\}$. That is a decision procedure, and does have a certificate: the subset of size $k$ is the certificate.
See also Optimization version of decision problems and standard textbooks.
