regarding the discussion question, my guess is that the only cases where the answer becomes unique are those 'trivial' single-vector or equivalent cases. More precisely those cases where min(nnz(row_sums),nnz(col_sums))<=1. The reasons is that if one solution exist where x(a,b)>0 and x(c,d)>0 (and a~=c and b~=d; i.e. they are not in the same row or column), then you could simply subtract one from x(a,b) and x(c,d), and add one to x(c,b) and x(a,d), and the result will also be a valid solution... thoughts?
I confess, I found the results using intlinprog.
I would love to know how you came up with that.
Counting in Finnish
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