It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard.

There is also a parameterized algorithm for computing a $\alpha$-approximation of VC (for $\alpha\in[1,2]$).

Considering the standard (non-parameterized) problem and non-polynomial algorithms:

  • What is the fastest known algorithm for computing a $1.99$-approximation?

I think for getting 1.99-approximation algorithm this paper by Manurangsi and Trevisan, has the current fastest algorithm.

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