Does the following approximation algorithm for vertex cover also have an approximation ratio of 2? Why? Why not?
Input: $G = (V,E)$
- Set $C \gets \emptyset$ and $E' \gets E$.
- while $E' \neq \emptyset$ do:
- Pick any edge $(u,v) \in E'$.
- $C \gets C \cup \{u\}$.
- Remove from $E'$ all edges incident to $u$.
- return $C$.