Google Research

On the Complexity of Fair House Allocation

Operations Research Letters, vol. 49 (4) (2021), pp. 572-577


We study fairness in house allocation, where m houses are to be allocated among n agents so that every agent receives one house. We show that maximizing the number of envy-free agents is hard to approximate to within a factor of n^{1−γ} for any constant γ > 0, and that the exact version is NP-hard even for binary utilities. Moreover, we prove that deciding whether a proportional allocation exists is computationally hard, whereas the corresponding problem for equitability can be solved efficiently.

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