Bipartite Graph - How to determine largest subsets that are all connected
Rephrasing from On Bipartite and Multipartite Clique Problems:

You can formulate the problem as an Integer Linear Program. Let the vertices in each partition be $V_1,V_2$ respectively. Take a (binary) variable $x_v$ for each $v\in V_1\cup V_2$. Then the problem is do determine

$$\textrm{maximize } \Sigma_{v\in V_1\cup V_2} x_v \textrm{ subject to}$$

$$x_v+x_u\leq 1 \textrm{ for all } u\in V_1, v\in V_2 \textrm{ so that } (u,v)\notin E$$

In general, solving ILP's is $NP$-complete. However, in this case, the corresponding constraint matrix is totally unimodular. Consider the relaxed problem: instead of requiring all $x_v$ to be binary, we instead only require that $0\leq x_v\leq 1$ (dropping the integrality constraint). Because the constraint matrix is totally unimodular, the optimal solution to the LP will be integral (and thus also a solution for the ILP). You can use the simplex method (or if you insist on wost-case polynomial time, the ellipsoid method) to obtain a solution for the LP, which is automatically a solution for the ILP.