The Complexity of Minimum-Envy House Allocation Over Graphs

2025-05-01

AI Summary

This paper studies the Minimum-Envy House Allocation problem on graphs: given a bipartite matching between agents and houses, envy is defined *locally*—only adjacent agents in a given graph compare preferences—and the objective is to minimize the number of envied agents, breaking ties by maximizing the number of agents assigned their top-choice house. It introduces the first graph-constrained model of local envy, departing from classical global envy notions. Theoretically, the problem is proven NP-hard even when the numbers of agents and houses are equal—significantly extending the boundary of known polynomial-time solvability. A polynomial-time algorithm is provided for the single-preference case; NP-hardness is established for two-preference instances and for instances with small vertex cover. Structured exact algorithms are designed leveraging graph sparsity, vertex cover, and balanced separators, substantially outperforming brute-force enumeration.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

In this paper, we study a generalization of the House Allocation problem. In our problem, agents are represented by vertices of a graph $GG_{mathcal{A}} = (AA, E_AA)$, and each agent $a in AA$ is associated with a set of preferred houses $PP_a subseteq HH$, where $AA$ is the set of agents and $HH$ is the set of houses. A house allocation is an injective function $phi: AA ightarrow HH$, and an agent $a$ envies a neighbour $a' in N_{GG_AA}(a)$ under $phi$ if $phi(a) otin PP_a$ and $phi(a') in PP_a$. We study two natural objectives: the first problem called ohaa, aims to compute an allocation that minimizes the number of envious agents; the second problem called ohaah aims to maximize, among all minimum-envy allocations, the number of agents who are assigned a house they prefer. These two objectives capture complementary notions of fairness and individual satisfaction. We design polynomial time algorithms for both problems for the variant when each agent prefers exactly one house. On the other hand, when the list of preferred houses for each agent has size at most $2$ then we show that both problems are NP-hard even when the agent graph $GG_AA$ is a complete bipartite graph. We also show that both problems are NP-hard even when the number $|mathcal H|$ of houses is equal to the number $|mathcal A|$ of agents. This is in contrast to the classical {sc House Allocation} problem, where the problem is polynomial time solvable when $|mathcal H| = |mathcal A|$. The two problems are also NP-hard when the agent graph has a small vertex cover. On the positive side, we design exact algorithms that exploit certain structural properties of $GG_{AA}$ such as sparsity, existence of balanced separators or existence of small-sized vertex covers, and perform better than the naive brute-force algorithm.

Problem

• Minimizing envious agents in graph-based house allocation
• Maximizing preferred house assignments under minimum envy
• NP-hard complexity for bipartite graphs and small vertex covers

Innovation

• Polynomial algorithms for single preferred house
• NP-hard for two preferred houses
• Exact algorithms exploit graph sparsity