For what values of A and B is the gap-VC-[A,B] problem NP-HARD?
Vertex cover has an easy $2$-approximation, so gap-VC-[$1/4,3/4$] is easy. The best unconditional hardness result is by Dinur and Safra, which gives $1.3606$ hardness. Since $(3/4)/(1/2) = 1.5 &gt; 1.3606$, their method definitely doesn't imply gap-VC-[$1/2,3/4$]-hardness. Assuming the Unique Games Conjecture, the hardness of vertex cover is $2-\epsilon$, but this doesn't necessarily imply that gap-VC-[$1/2,3/4$] is hard - you'd have to look at the proof to determine that.

In summary, I have no idea what the correct answer is, and I'm not sure that anyone else knows which answer is correct (unconditionally).