Bipartite Matching

I need to find an algorithm (preferably in Java) to solve the following problem (hoping it will be clearly expressed):

Given a matrix (not necessarily square) of 1 and 0 values, like the following:

I must be able to determine the maximum number of cells, so that there are no pairs of cells among those selected having a row or a column in common.

For example, if the cell (Row_A, Col_Y) was selected, then the cells (Row_A, Col_V), (Row_A, Col_S), (Row_C, Col_Y), (Row_G, Col_Y) must be excluded.

The problem must be tackled as a bipartite graph, where a partition of the nodes is repres开发者_如何学Cented by rows, and columns from the other. There is a link only between nodes that have 1 in their respective cells.

So we will have the partition Part_Row, that will contain the following nodes: A, B, C, D, E, F, G. While the partition Part_Col will contain the nodes: Z, Y, X, W, V, T, S, R, Q. The arches will be:

A->Y, A-​​>S
B->Z, B->D
C->Y, C->X, C->S,
etc., etc.

How can I determine the maximum number of cells? Does it make sense to solve the maximum matching problem as a problem of maximum flow?