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[Solved]: For what values of A and B is the gap-VC-[A,B] problem NP-HARD?

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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.

Asked By : Jozef

Answered By : Yuval Filmus

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 $(3/4)/(1/2) = 1.5 > 1.3606$, 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).

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