The Cost and Complexity of Minimizing Envy in House Allocation
Jayakrishnan Madathil, Neeldhara Misra, Aditi Sethia
TL;DR
The paper addresses three notions of envy minimization in one-sided house allocation: minimizing the number of envious agents (OHA), the maximum envy experienced by any agent (EHA), and the total envy across all agents (UHA). It develops a broad set of algorithmic and complexity results, including polynomial-time algorithms for several structured regimes (notably extremal interval valuations and $m=n$), hardness results (NP-hard and para-NP-hard under various restrictions), and kernelization via the Expansion Lemma to obtain kernels with $|H|\le 2(|A|-1)$. It provides ILP formulations and FPT results parameterized by agent/house types, and explores the price of fairness to quantify welfare loss when enforcing fairness. The experimental evaluation of ILPs demonstrates practical performance in synthetic settings and highlights how agent diversity and house availability influence envy and welfare, offering insights into the cost of fairness in real-world allocations.
Abstract
We study almost-envy-freeness in house allocation, where $m$ houses are to be allocated among $n$ agents so that every agent receives exactly one house. An envy-free allocation need not exist, and therefore we may have to settle for relaxations of envy-freeness. But typical relaxations such as envy-free up to one good do not make sense for house allocation, as every agent is required to receive exactly one house. Hence we turn to different aggregate measures of envy as markers of fairness. In particular, we define the amount of envy experienced by an agent $a$ w.r.t. an allocation to be the number of agents that agent $a$ envies under that allocation. We quantify the envy generated by an allocation using three different metrics: 1) the number of agents who are envious; 2) the maximum amount of envy experienced by any agent; and 3) the total amount of envy experienced by all agents, and look for allocations that minimize one of the three metrics. We thus study three computational problems corresponding to each of the three metrics and prove a host of algorithmic and hardness results. We also suggest practical approaches for these problems via integer linear program (ILP) formulations and report the findings of our experimental evaluation of ILPs. Finally, we study the price of fairness (PoF), which quantifies the loss of welfare we must suffer due to the fairness requirements, and we prove a number of results on PoF, including tight bounds as well as algorithms that simultaneously optimize both welfare and fairness.