Given a random matrix $R$ whose entries are either $0$ or $1$. Does there exist an efficient algorithm that can tackle the following problem rather than brute force or is there any keyword for this?
$$\mathop{\arg\max}_\limits{S} \ \left | S \right | \quad \text{subject to} \quad R_{ij} = 1 \, , \ \forall \, i,j \in S $$
$S$ is a set storing part of the given random matrix indices, and the constraint is that each element whose numbers of row and column are in $S$ is $1$. Besides, the solution of $S$ may be more than one.