For which values $A,B$ is the problem $\mathsf{gap\mathord-VC}\mathord-[A,B]$ NP-hard? VC is the vertex cover problem. I am given three options: $B=\frac{3}{4},A=\frac{1}{2}$ or $B=\frac{3}{4},A=\frac{1}{4}$ or none.
I would to review what I think that I need to do, I'm not sure that the way I think of it is correct. This is what I think: I need to decide if it NP-hard to approximate the VC to $\frac{1}{2}$, i.e., can I build an NP Turing machine that would return Yes iff for a given graph, it can guarantee that it has less than $\frac{1}{4}V$ vertices that cover the whole graph? Maybe even for $\frac{1}{2}V$ vertices?
This is a question from a past midterm that I'm solving now in order to prepare myself for my own midterm in a "Computational Complexity Theory" course.