Define an independent set of a graph $G = (V, E)$ to be a subset $S$ of vertices such that $V-S$ is a vertex cover of $G$. Is every $2$-approximation algorithm for finding a minimum vertex cover also a $2$-approximation algorithm for finding a maximum independent set?
I have read about approximation algorithms, but could relate vertex cover problem to independent set. Do you have any suggestions on why such algorithm can be found or not?
Well as far as I am concerned, unless P=NP there can't be a constant-ratio approximation algorithm for the Maximum Independent set problem On the Complexity of Approximating the independent Set Problem . So to answer your question:
No.
A 2-approximation algorithm for the Minimum Vertex Cover does not guarantee for an Independent set having at least half the size of the Maximum Independent set. Suppose $S$ is an optimal vertex cover and has size $\geq |V|/2$. Then a 2-approximation algorithm is only guaranteed to give a vertex cover of size $|V|$. Hence one does not obtain a non-trivial independent set by complementing the approximate vertex cover.