smaller size approximation to minimum vertex cover
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.