@T.Amdeberhan : In the problem mentioned by you(mathoverflow.net/questions/268798/…) the size of subset is fixed. Here my subsets can be of any size. Though by the constraints you can prove that each of them have to be of atleast size 2.

But the thing is i $\in$ B(i) is not allowed. That is the additional constraint. In graph terms you cannot have self loops.

The dual problem of this seems to be messy. If you look at it from the graph theory perspective I am claiming that when the graph has only directed edge from i to j then it cannot have an edge from j to i. Also after constructing such subsets for all 1,2,...,n ; this condition would never get violated. This seems to be difficult considering the subset constraints that are present. Please let me know if you get better insights on it. If possible can you tell me what you think is a dual problem for this.

@Fedor: Could you elaborate on it or give me some link for reference.