Your guess is absolutely correct. Consider vertical and horizontal segments formed by adjacent symbols 'A' in every row and in every column. If we want to place a "fighter" in some square (i, j), we can't place any "fighter" in vertical and in horizontal segments that (i,j) belongs to.

So, we need to make a bipartite graph where vertices are segments and edges are squares (i,j) where we can place a "fighter".

Maximal matching in this graph is the answer for the problem.