In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable partition always does and provides a succinct certificate for the unsolvability of an instance. Furthermore, apart from being a useful structural tool to study the problem, every stable partition corresponds to a stable half-matching, which has applications, for example, in sports scheduling and time-sharing. We establish new structural results for stable partitions and show how to enumerate all stable partitions and the cycles included in such structures efficiently. We also adapt optimality criteria from stable matchings to stable partitions and give complexity and approximability results for the problems of computing such "fair" and "optimal" stable partitions. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial point of view, as well as the computational complexity of computing "fair" or "optimal" stable half-matchings in practice, closing the gap between integral and fractional stable matchings and paving the way for further applications of stable partitions to unsolvable instances and computationally hard stable matching problems.

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