Graphical House Allocation

The classical house allocation problem involves assigning 𝑛 houses (or items) to𝑛 agents according to their preferences. A key criteria in such problems is satisfying some fairness constraints such as envyfreeness. 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 the envy value 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; we also obtain 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.

Authors

• Hadi Hosseini
• Justin Payan University of Massachusetts, Amherst
• Rik Sengupta UMass Amherst
• Rohit Vaish Indian Institute of Technology Delhi
• Vignesh Viswanathan University of Massachusetts Amherst

Keywords

• Fair Allocation
• House Allocation
• Envy Minimization