I am interested in the complexity of the restricted version of the vertex cover problem below:
Instance: A bipartite graph $G =(L, R, E)$ and an integer $K$.
Question: Is there $S \subset L$, $|S| \leq K$ and every vertex in $R$ has a neighbor in $S$ $( S$ is vertex cover for $R)$
Vertex cover is $\mathsf{P}$ if $S \subset L \cup R$ and cover $L \cup R$; and it is $\mathsf{NP}$-complete for nonbipartite graphs. However, the problem I am looking at does not fit in either cases. Any pointers where I could find an answer will be appreciated.