It is a many-to-one assignment problem with N tasks and M people.
Each person can get multiple tasks, while each task can be assigned to only one person. We can earn a profit Pij if the task i is assigned to person j.
If T1, T2,..., Tm is a partition of the tasks, and n1, n2,..., nm are m positive integers. I want the optimum assignment such that the number of people assigned to any task in Ti must be less or equal to ni
If I understand your question correctly, this is a special case of the minimum-cost flow problem on a graph with three layers (in addition to a source and a sink layer).
We haven't specified a demand, but we could simply try all possible demands between 0 and M and still be in P , so showing that it's not NP -hard is equivalent to showing that P ≠ NP .
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