I am trying to find a combinatorial approach to solve the following optimization problem. 

\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\forall i \in [N],\\
&\sum_{i} x_{ij} \leq c_j~\forall j \in [M],\\
&x_{ij} \geq 0~\forall i,j,
\end{align}
where the constants satisfy: $C_{ij} \geq 0$, $\sum_i r_i = 1$. 

If $\sum_j c_j = 1$ then, I think, this problem is similar to the discrete (finite) optimal transport problem. 

I am not interested in the most efficient solution approach. I am interested in an approach that reveals any interesting structure of a solution if such a structure exists. 

In particular, is there a greedy algorithm (after sorting the weights $C_{ij}$) that solves this problem?