We select one edge after the other and we check whether the extremes of this edge are already covered. If the extremes are not in the vertex cover, then we add to the vertex cover both extremes of the edge.
This is very simple, but still very powerful because it is possible to prove that this algorithm computes in polynomial-time a 2-approximation solution.
A prove of these results is based in the following observations. If you think about this algorithm, the resulting solution is basically form by a matching, that is a collection of edges, which do not intercept with each other. This is called the matching and the result solution of this algorithm is a matching. Since each edge of this matching has to be covered, it means that any solution has to take at least one node for each of these edges.
Our solution takes 2 nodes for each of these edges, so this means that our solution is at most twice the minimum solution.
