Minimum vertices set bipartite graph covering-special case
Solution 1:
This is just the set cover problem in disguise, to each vertex of $L$ assign the set of vertices of $R$ in its neighborhood. We now have transformed $L$ into a set of subsets of $R$ so now we want to find the set covering with the fewest sets.
It is true, as you suspected that this problem is $NP$ complete, but a lot of research on it exists.