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 $n^{1-\gamma}$ for any constant $\gamma>0$, 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.

中文翻译：

---

公平房屋分配的复杂性

我们研究了房屋分配的公平性，其中 $m$ 的房屋将分配给 $n$ 的代理人，以便每个代理人获得一所房子。我们表明，对于任何常数 $\gamma>0$ 而言，最大化无嫉妒代理的数量很难接近于 $n^{1-\gamma}$ 的因子，并且确切的版本是 NP-hard 甚至用于二进制实用程序。此外，我们证明决定是否存在比例分配在计算上是困难的，而公平性的相应问题可以有效地解决。