A Minimax Perspective on Almost-Stable Matchings

2026-01-20

AI Summary

In real-world settings, perfectly stable matchings are often unattainable, making the fair distribution of instability a critical concern. This work proposes a novel notion of approximate stability from a minimax perspective, aiming to minimize the maximum number of blocking pairs incident to any individual, thereby ensuring fairness for the worst-off participant. The study investigates this concept within both the Stable Marriage and Stable Roommates frameworks, combining computational complexity analysis, polynomial-time algorithms, integer programming, and approximation techniques. Key contributions include proving that the problem remains NP-complete even when each agent is allowed to belong to at most one blocking pair, designing efficient algorithms for instances where preference lists have length at most two, and providing effective approximation and integer programming formulations for practical computation.

Technology Category

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

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

Abstract

Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either impossible to achieve or can come at the cost of leaving many agents unmatched. When stability cannot be achieved, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to"almost-stable"matchings focus on aggregate measures, minimising either the total number of blocking pairs or the count of agents involved in blocking pairs. However, such aggregate objectives can result in concentrated instability on a few individual agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists have constant-bounded length. This hardness applies to both Stable Roommates and maximum-cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees and computational feasibility.

Problem

• stable matching
• almost-stable matching
• minimax fairness
• blocking pairs
• computational complexity

Innovation

• minimax fairness
• almost-stable matching
• blocking pairs
• computational complexity
• stable roommates