Task allocation problems have traditionally focused on cost optimization. However, more and more attention is being given to cases in which cost should not always be the sole or major consideration. In this paper, we study a fair task allocation problem where an optimal allocation not only has low cost but more importantly, it is max-lexmin fair to all participants. To tackle this max-lexmin fair minimum cost allocation problem, we analyze and solve it in two parts. We first aim to determine the maximum number of jobs that can be feasibly done and the fairest distribution thereof. Since there may be many allocations that are considered equally fair, we then find the allocation with the minimum costs. We propose two novel polynomial-time algorithms that make use of the network flow structure of the problem to obtain the optimal solution. Furthermore, we conduct extensive experiments to investigate the trade-off between cost minimization and fairness.
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