Perspectives on Unsolvability in the Roommates Problem

TL;DR

This paper analyzes the probability and structural factors influencing solvability in the Stable Roommates problem, combining theoretical, probabilistic, and experimental methods to understand when stable matchings exist and their characteristics.

Contribution

It provides the first comprehensive empirical and structural analysis of unsolvability in the Stable Roommates problem, including estimates of solvability probability and stable solution set sizes.

Findings

Probability of solvability is low for most distributions.

Stable solution sets are typically small.

Many related NP-hard problems are practically tractable.

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.