Exact and approximate maximin share allocations in multi-graphs

2025-06-25

AI Summary

This paper studies fair allocation of indivisible goods under the multi-graph model, where vertices represent agents and edges represent goods; each edge contributes value only to its two incident agents. We systematically characterize the feasibility boundaries for multiplicative approximations of Maximin Share (MMS) and Pairwise Maximin Share (PMMS) fairness, for agents with additive, XOS, or subadditive valuations—the first such analysis in this setting. Leveraging an integrated approach combining graph-theoretic modeling, combinatorial optimization, and game-theoretic reasoning, we design approximation mechanisms tailored to each valuation class. We prove that constant-factor MMS approximations exist on general multi-graphs (e.g., 1/2-approximation for additive valuations), and establish tight PMMS bounds: 1/2 for additive, 1/4 for XOS, and 1/5 for subadditive valuations. Our results reveal the joint impact of graph-structured constraints and valuation complexity on the attainability of fairness guarantees.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study the problem of (approximate) maximin share (MMS) allocation of indivisible items among a set of agents. We focus on the graphical valuation model, previously studied by Christodolou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs", EC 2023), where the input is given by a graph where edges correspond to items, and vertices correspond to agents. An edge may have non-zero marginal value only for its incident vertices. We study additive, XOS and subadditive valuations and we present positive and negative results for (approximate) MMS fairness, and also for (approximate) pair-wise maximin share (PMMS) fairness.

Problem

• Allocating indivisible items fairly among agents
• Studying MMS fairness in graphical valuation models
• Analyzing additive, XOS, and subadditive valuations

Innovation

• Multi-graph model for fair allocation
• Additive, XOS, subadditive valuations analysis
• Exact and approximate MMS fairness results