Is it possible to compute the minimum vertex covering set in quasi-polynomial time, by knowing the vertex cover number?
Suppose there is an algorithm $A(G,c)$ which runs in time $T(n)$, where $n$ is the number of vertices in $G$, as long as $c$ is the vertex cover number.
Given a graph $G$, run $A(G,0),\ldots,A(G,n)$ in parallel, for $T(n)$ time each. Some of the copies will terminate within $T(n)$, outputting a set. Out of all such outputs which are vertex covers, choose the one of minimal size. This gives an algorithm for minimum vertex cover running in time $O(nT(n))$.