On the Complexity of Fair House Allocation

TL;DR

This paper investigates the computational complexity of achieving fair house allocations, demonstrating NP-hardness and inapproximability results for envy-freeness and proportionality, while showing that equitability can be efficiently decided.

Contribution

It establishes new hardness results for maximizing envy-free agents and deciding proportional allocations, and identifies cases where equitable allocations are efficiently computable.

Findings

Maximizing envy-free agents is hard to approximate within n^{1-γ}.

Deciding the existence of proportional allocations is NP-hard.

Equitability can be decided efficiently.

Abstract

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.