An instance of the Stable Roommates problem involves a set of agents, each with ordinal preferences over the others. We seek a stable matching, 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 for instances with a large number of agents. However, stable partitions always exist and provide a succinct certificate for the unsolvability of an instance, although their significance extends beyond this. They are also a useful structural tool to study the problem and correspond to half-matchings in which the agents are in a stable equilibrium. In this paper, we investigate the stable partition structure further and show how to efficiently enumerate all stable partitions and the cycles included in such structures. Furthermore, we adapt known fairness and optimality criteria from stable matchings to stable partitions. As there can be an exponential number of stable partitions, we investigate the complexity of computing different "fair" and "optimal" stable partitions directly. While a minimum-regret stable partition always exists and can be computed in linear time, we prove the NP-hardness of finding five other kinds of stable partitions that are "optimal" regarding their profile (measuring the number of first, second, third, etc., choices assigned). Furthermore, we give 2-approximation algorithms for two of the optimal stable partition problems and show the inapproximability within any constant factor for another. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial and complexity point of view, closing the gap between integral and fractional stable matchings.

翻译：稳定室友问题的一个实例涉及一组智能体，每个智能体对其他智能体具有序数偏好。我们寻求一种稳定匹配，其中任意两个智能体都没有动机偏离其分配。众所周知，对于具有大量智能体的实例，稳定匹配很可能不存在。然而，稳定划分总是存在，并为实例的不可解性提供了简洁的证明，尽管其意义远不止于此。它们也是研究该问题的有用结构工具，并对应于智能体处于稳定均衡状态的半匹配。本文中，我们进一步研究了稳定划分结构，展示了如何高效枚举所有稳定划分以及此类结构中所包含的圈。此外，我们将稳定匹配中已知的公平性和最优性标准适配到稳定划分上。由于稳定划分的数量可能是指数级的，我们研究了直接计算不同“公平”和“最优”稳定划分的复杂度。虽然最小遗憾稳定划分总是存在且可在线性时间内计算，但我们证明了寻找其他五种在偏好剖面（衡量所分配的第一、第二、第三等选择的数量）方面“最优”的稳定划分是NP难的。此外，我们针对其中两个最优稳定划分问题给出了2-近似算法，并证明了另一个问题无法在任意常数因子内近似。通过这项研究，我们从组合和复杂度的角度促进了对稳定划分的更深入理解，从而弥合了整数稳定匹配与分数稳定匹配之间的差距。