Graphical House Allocation

TL;DR

This paper studies a generalized graph-based house allocation problem aiming to minimize envy among agents connected in a social network, providing structural insights and algorithms for various graph classes.

Contribution

It introduces a novel graph-based fairness model for house allocation, analyzes its complexity, and develops algorithms for specific graph classes.

Findings

NP-hardness for disjoint unions of paths, cycles, stars, or cliques

Fixed-parameter tractable algorithms for paths, cycles, stars, cliques

Polynomial-time algorithms for some graph classes

Abstract

The classical house allocation problem involves assigning $n$ houses (or items) to $n$ agents according to their preferences. A key criterion in such problems is satisfying some fairness constraints such as envy-freeness. We consider a generalization of this problem wherein the agents are placed along the vertices of a graph (corresponding to a social network), and each agent can only experience envy towards its neighbors. Our goal is to minimize the aggregate envy among the agents as a natural fairness objective, i.e., the sum of all pairwise envy values over all edges in a social graph. When agents have identical and evenly-spaced valuations, our problem reduces to the well-studied problem of linear arrangements. For identical valuations with possibly uneven spacing, we show a number of deep and surprising ways in which our setting is a departure from this classical problem. More broadly, we contribute several structural and computational results for various classes of graphs, including NP-hardness results for disjoint unions of paths, cycles, stars, or cliques, and fixed-parameter tractable (and, in some cases, polynomial-time) algorithms for paths, cycles, stars, cliques, and their disjoint unions. Additionally, a conceptual contribution of our work is the formulation of a structural property for disconnected graphs that we call separability which results in efficient parameterized algorithms for finding optimal allocations.