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? 
What is the Certificate for Set Cover?
complexity theory;optimization