Minimum Envy Graphical House Allocation Beyond Identical Valuations

2026-01-22

AI Summary

This study addresses the graphical house allocation problem under non-identical valuations, where a social graph encodes agent relationships and agents hold heterogeneous preferences over houses. The objective is to compute an allocation that minimizes total envy measured along the edges of the graph. By jointly exploiting structural properties of the graph—such as maximum degree at most one—and valuation characteristics—such as binary preferences—the authors develop polynomial-time exact algorithms, approximation algorithms, and moderately exponential-time algorithms. This work presents the first systematic analysis of how graph topology and valuation heterogeneity jointly influence computational complexity. Efficient solutions are achieved for several graph classes, including paths and matchings; notably, when the graph has maximum degree one and valuations are binary, an allocation with total envy at most one can be found in polynomial time.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

House allocation is an extremely well-studied problem in the field of fair allocation, where the goal is to assign $n$ houses to $n$ agents while satisfying certain fairness criterion, e.g., envy-freeness. To model social interactions, the Graphical House Allocation framework introduces a social graph $G$, in which each vertex corresponds to an agent, and an edge $(u, v)$ corresponds to the potential of agent $u$ to envy the agent $v$, based on their allocations and valuations. In undirected social graphs, the potential for envy is in both the directions. In the Minimum Envy Graphical House Allocation (ME-GHA) problem, given a set of $n$ agents, $n$ houses, a social graph, and agent's valuation functions, the goal is to find an allocation that minimizes the total envy summed up over all the edges of $G$. Recent work, [Hosseini et al., AAMAS 2023, AAMAS 2024] studied ME-GHA in the regime of polynomial-time algorithms, and designed exact and approximation algorithms, for certain graph classes under identical agent valuations. We initiate the study of \gha with non-identical valuations, a setting that has so far remained unexplored. We investigate the multivariate (parameterized) complexity of \gha by identifying structural restrictions on the social graph and valuation functions that yield tractability. We also design moderately exponential-time algorithms for several graph classes, and a polynomial-time algorithm for {binary valuations that returns an allocation with envy at most one when the social graph has maximum degree at most one.

Problem

• Graphical House Allocation
• Minimum Envy
• Non-identical Valuations
• Fair Allocation
• Envy-freeness

Innovation

• Graphical House Allocation
• non-identical valuations
• parameterized complexity
• minimum envy
• binary valuations