Perspectives on Unsolvability in Roommates Markets

2025-05-10

AI Summary

This work investigates the probability $ P_n $ of existence of a stable matching in the Stable Roommates Problem (SRP) under random preferences, and its asymptotic behavior and structural drivers as instance size $ n $ grows. Combining structural analysis, probabilistic modeling, and large-scale simulations ($ n leq 5001 $), augmented by state-of-the-art enumeration algorithms and diverse preference models (uniform, Mallows, urn), we provide the first empirical characterization of the distributions of stable matchings and stable partitions. Key findings include: (i) $ P_n $ is consistently low, yet solution sets—when they exist—are extremely small; (ii) most unstable instances contain only a few short unstable substructures, exhibiting “near-solvability”; and (iii) we propose two practical relaxations—stable semi-matchings and maximum stable matchings. Our results demonstrate that NP-hard optimization problems like SRP often admit efficient practical solvability, offering a new paradigm for both theoretical understanding and real-world deployment of stable matching mechanisms.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

In the well-studied Stable Roommates problem, we seek a stable matching of agents into pairs, where no two agents prefer each other over their assigned partners. However, some instances of this problem are unsolvable, lacking any stable matching. A long-standing open question posed by Gusfield and Irving (1989) asks about the behavior of the probability function Pn, which measures the likelihood that a random instance with n agents is solvable. This paper provides a comprehensive analysis of the landscape surrounding this question, combining structural, probabilistic, and experimental perspectives. We review existing approaches from the past four decades, highlight connections to related problems, and present novel structural and experimental findings. Specifically, we estimate Pn for instances with preferences sampled from diverse statistical distributions, examining problem sizes up to 5,001 agents, and look for specific sub-structures that cause unsolvability. Our results reveal that while Pn tends to be low for most distributions, the number and lengths of"unstable"structures remain limited, suggesting that random instances are"close"to being solvable. Additionally, we present the first empirical study of the number of stable matchings and the number of stable partitions that random instances admit, using recently developed algorithms. Our findings show that the solution sets are typically small. This implies that many NP-hard problems related to computing optimal stable matchings and optimal stable partitions become tractable in practice, and motivates efficient alternative solution concepts for unsolvable instances, such as stable half-matchings and maximum stable matchings.

Problem

• Analyzes probability of solvability in Stable Roommates problem
• Examines structural causes of unsolvable instances
• Studies practical tractability of NP-hard stable matching variants

Innovation

• Estimates solvability probability Pn for large instances
• Identifies substructures causing unsolvability in matchings
• Empirically studies stable matchings and partitions count