What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

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?