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 n1−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.
@article{arxiv.2106.06925,
title = {On the Complexity of Fair House Allocation},
author = {Naoyuki Kamiyama and Pasin Manurangsi and Warut Suksompong},
journal= {arXiv preprint arXiv:2106.06925},
year = {2021}
}