I am currently dealing with a network flow problem and I am trying to find some similar solved problems to help me formulate my solution.
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You are the owner of a large chain of franchise shops, and you would like to expand to a new city. The blocks in your city make an $n ×n$ grid. However, although your products are awesome and in high demand, the city will not allow you to open a shop in each block. Instead, for every row of blocks i, you are are given a number $r_i$ that limits the maximum number of shops opened there - and for every column j, there also is a maximum number $c_j$ .
a) Find the maximum number of franchise shops you can legally open in the city. To do so, model the problem as a flow network. Then, describe how to get the right answer using Ford-Fulkerson, and prove the correctness of your construction.
How can I construct the digraph from the $n ×n$ grid ? I think the blocks can be represented in this way grid Should I assume an initial orientation on the grid?