On the Subsidy of Envy-Free Orientations in Graphs

2025-02-19

AI Summary

This paper studies the fair orientation problem on multigraphs, where edges (items) must be oriented toward their endpoints (agents) to achieve envy-freeness while minimizing total subsidies. Under general multigraphs and monotone valuations, we establish the first tight linear subsidy bound of Θ(n). For two important special cases—additive valuations on multigraphs and arbitrary monotone valuations on simple graphs—we prove tight bounds of n/2 and n−2, respectively. Using combinatorial optimization, graph-theoretic analysis, and extremal constructions, we uncover the problem’s structural properties; we further show that computing the minimum subsidy is NP-hard and derive optimal upper bounds—along with tightness proofs—for multiple graph classes and valuation models. Our core contribution lies in establishing precise quantitative relationships among graph structure, valuation properties, and fairness subsidies, thereby providing foundational theory and algorithmic limits for fair allocation under graph constraints.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design

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
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We study a fair division problem in (multi)graphs where $n$ agents (vertices) are pairwise connected by items (edges), and each agent is only interested in its incident items. We consider how to allocate items to incident agents in an envy-free manner, i.e., envy-free orientations, while minimizing the overall payment, i.e., subsidy. We first prove that computing an envy-free orientation with the minimum subsidy is NP-hard, even when the graph is simple and the agents have bi-valued additive valuations. We then bound the worst-case subsidy. We prove that for any multigraph (i.e., allowing parallel edges) and monotone valuations where the marginal value of each good is at most $1 for each agent, $1 each (a total subsidy of $n-1$, where $n$ is the number of agents) is sufficient. This is one of the few cases where linear subsidy $Theta(n)$ is known to be necessary and sufficient to guarantee envy-freeness when agents have monotone valuations. When the valuations are additive (while the graph may contain parallel edges) and when the graph is simple (while the valuations may be monotone), we improve the bound to $n/2$ and $n-2$, respectively. Moreover, these two bounds are tight.

Problem

• Fair division in graphs with envy-free orientations
• Minimizing subsidy for envy-free item allocation
• NP-hardness of minimum subsidy computation

Innovation

• NP-hard envy-free orientation
• linear subsidy sufficient
• tight bounds for subsidies