Suppose we have $n$ workers and $m$ tasks. Suppose each task has a workforce $S_i \subseteq \{1, 2, .., n\}$. Note that $S_i \cap S_j \neq \emptyset$ is possible (and expected).
Given this configuration, we want to assign a leader $s_i$ for each task $i$ such that $s_i \in S_i$.
Moreover, we want a fairness criterion for each person:
If a person is in taskforces $S_{i_1}, ..., S_{i_k}$ with sizes $n_1, ..., n_k$ respectively, then the person should be given at most $\frac{1}{n_1} + \frac{1}{n_2} + ... + \frac{1}{n_k}$ tasks to lead or as this may not be an integer, the person should be given at most $\lceil\frac{1}{n_1} + \frac{1}{n_2} + ... + \frac{1}{n_k}\rceil$ many tasks to lead
Is there a polynomial time algorithm that, given taskforces $S_1, ..., S_m$ assigns leaders in a fair way (where "fairness" is determined as above)?
I think we have to write this as a max flow problem and use algorithms such as Edmonds & Karp, Dinics, etc. Not sure though.