A matrix decomposition

Folks,

I've got a problem, and a working solution, but I've got a feeling
this might be a well-studied case that I am unaware of.

I've got a n x n matrix A with a zero diagonal, and whose entries
are 0 or 1. I need to express A as a sum of matrices

   A = A_1 + A_2 + ... + A_m

such that each A_k has row sums and column sums that are bounded
above by 1. In other words, each row and column of A_k can have at
most one nonzero entry.

I am looking for such a decomposition that has the smallest number of
matrices. It is clear that m is bounded above by the number of nonzero
entries in A and bounded below by the maximum row or column sum of A.

For most of my applications, A is sparse but not too sparse, and the
decomposition I find has close to the lower bound of terms. It would
be of interest to find the minimum number of terms, and an algorithm
to find such a decomposition with this minimum number of terms.

As a generalization, I also have A with a zero diagonal and whose
offdiagonal entries are nonnegative (not just 0 or 1), and I need 
to find such a decomposition with close to a minimum number of terms.

Any insight I would appreciate.

--Cleve Ashcraft