smaller size approximation to minimum vertex cover Does there exist a simple approximation to the minimum vertex cover problem that aims to find a smaller (or equal) set to the minimum?

Usual algorithms seems to aim to find an approximation such that the output is a cover, but may have more vertices than the minimum cover. Instead, I want a smaller set - I don't mind if some edges are still left and is therefore not a cover. Between two smaller sets, the one that covers more (total number of edges) is preferable.
Do you want a set whose size is a lower bound? Or do you want a lower bound on the size?

If you want a set, does it need to be a subset of a minimum vertex cover, or just a smaller set (in the latter case, any set of the appropriate cardinality would do)?

Since there is a well-known approximation algorithm that is a factor of two over-approximation, you can divide the result by two to get a factor of two under-approximation. In fact, these notes use a lower bound to prove the over-approximation factor.