Given a graph $G=(V,E)$, and two positive integers $k$ and $\gamma$, decide if there is a set of new edges to be added such that $|E'|=k$, $E' \cap E = \emptyset$ and any subset $V'\subseteq V$ of size $\gamma$ is not a vertex cover of $(V, E\cup E')$.
i.e., can we add $k$ edges to ensures that the minimum vertex cover has a size at least $\gamma+1$.
Clearly, this problem is co-$NP$-hard, even for $k=0$, as this is then the negation of the regular vertex-cover problem. But for general $k$, does this problem allow any approximation algorithms?
