Vertex Cover - Understanding the bounds

Question

I was reading on wikipedia about the approximations of the Vertex Cover problem and saw that an approximation algorithm with an approximation factor of $\displaystyle 2 - \Theta \left( \frac{1}{\sqrt{\log |V|}} \right)$ is known.

I want to understand how to read that expression.

Any help small or big would be appreciated.

Answer 1

lets say $f(|V|)$ is the real approximation factor. then $f(|V|) = 2-\Theta \left( \frac{1}{\sqrt{\log |V|}} \right ) $ means the same as $2-f(|V|) = \Theta \left ( \frac{1}{\sqrt{\log |V|}} \right ) $ which means that $\frac{k_2}{\sqrt{\log |V|}} \leq 2-f(|V|) \leq \frac{k_1}{\sqrt{\log |V|}} $ for some constant $k_1,k_2$

(source: https://en.wikipedia.org/wiki/Big_O_notation#Related_asymptotic_notations)

Answer 2

Without reading the article myself:

I assume this means that in the limit, the approximation factor is 2, and it approaches $2$ with a function that is in $$\Theta\left(\frac{1}{\sqrt{log|V|}}\right)$$

Or in other words, the smaller the graph, the better the approximation, but never worse than $2$ (i assume).