Envy-Free House Allocation with Minimum Subsidy

TL;DR

It is shown that computing an envy-free allocation with minimum subsidy is NP-hard in general, but can be done efficiently if $m$ differs from $n$ by an additive constant or if the agents have identical utilities.

Abstract

House allocation refers to the problem where $m$ houses are to be allocated to $n$ agents so that each agent receives one house. Since an envy-free house allocation does not always exist, we consider finding such an allocation in the presence of subsidy. We show that computing an envy-free allocation with minimum subsidy is NP-hard in general, but can be done efficiently if $m$ differs from $n$ by an additive constant or if the agents have identical utilities.

Theorems & Definitions (15)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Theorem 3.1
• proof
• Theorem 4.1
• Proposition 4.2
• Lemma 4.3: halpern2019fair
• Lemma 4.4: Lemma 3 of barman2022achieving
• Lemma 4.5