# On random stable partitions

## TL;DR

This paper investigates the properties of stable partitions in the random stable roommates problem, revealing growth rates and probabilities related to odd cycles, stable matchings, and blocking pairs as the number of members increases.

## Contribution

It provides asymptotic estimates for the expected number of stable partitions with odd cycles and related structural properties in the random stable roommates problem.

## Key findings

- Expected number of stable partitions with odd cycles grows as n^{1/4}
- Fraction of members with multiple predecessors is of order n^{-1/4}
- Largest stable matching size is approximately n/2 minus a term of order n^{1/4}

## Abstract

The stable roommates problem does not necessarily have a solution, i.e. a stable matching. We had found that, for the uniformly random instance, the expected number of solutions converges to $e^{1/2}$ as $n$, the number of members, grows, and with Rob Irving we proved that the limiting probability of solvability is $e^{1/2}/2$, at most. Stephan Mertens's extensive numerics compelled him to conjecture that this probability is of order $n^{-1/4}$. Jimmy Tan introduced a notion of a stable cyclic partition, and proved existence of such a partition for every system of members' preferences, discovering that presence of odd cycles in a stable partition is equivalent to absence of a stable matching. In this paper we show that the expected number of stable partitions with odd cycles grows as $n^{1/4}$. However the standard deviation of that number is of order $n^{3/8}\gg n^{1/4}$, too large to conclude that the odd cycles exist with high probability (whp). Still, as a byproduct, we show that whp the fraction of members with more than one stable "predecessor" is of order $n^{-1/4}$. Furthermore, whp the average rank of a predecessor in every stable partition is of order $n^{1/2}$. The likely size of the largest stable matching is $n/2-O(n^{1/4+o(1)})$, and the likely number of pairs of unmatched members blocking the optimal complete matching is $O(n^{3/4+o(1)})$.