Beating the Logarithmic Barrier for the Subadditive Maximin Share Problem

2025-06-05

AI Summary

This paper studies fair allocation of indivisible goods under subadditive valuations, aiming to improve the maximin share (MMS) guarantee. Prior work achieved only a $1/O(log n log log n)$ approximation ratio—the best known—leaving a significant gap in fairness guarantees. We break this barrier by establishing, for the first time, the existence of a $1/O((log log n)^2)$-approximate MMS allocation for subadditive agents, yielding the strongest MMS approximation ratio to date. Technically, we introduce a novel *matching-and-rounding* framework that integrates combinatorial optimization, tailored bipartite matching, iterative rounding, and localized correction. This framework overcomes fundamental challenges arising from valuation nonlinearity and interdependence inherent in subadditivity. Our result establishes a new theoretical benchmark for fair division under subadditive preferences and provides a foundational algorithmic tool for future research.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the problem of fair allocation of indivisible goods for subadditive agents. While constant- extsf{MMS} bounds have been given for additive and fractionally subadditive agents, the best existential bound for the case of subadditive agents is $1/O(log n log log n)$. In this work, we improve this bound to a $1/O((log log n)^2)$- extsf{MMS} guarantee. To this end, we introduce new matching techniques and rounding methods for subadditive valuations that we believe are of independent interest and will find their applications in future work.

Problem

• Fair allocation of indivisible goods for subadditive agents
• Improving MMS guarantee for subadditive valuations
• Developing new matching and rounding techniques

Innovation

• Improved MMS guarantee for subadditive agents
• New matching techniques for valuations
• Advanced rounding methods for allocations