In the fundamental Stable Marriage and Stable Roommates problems, there are inherent trade-offs between the size and stability of solutions. While in the former problem, a stable matching always exists and can be found efficiently using the celebrated Gale-Shapley algorithm, the existence of a stable matching is not guaranteed in the latter problem, but can be determined efficiently using Irving's algorithm. However, the computation of matchings that minimise the instability, either due to the presence of additional constraints on the size of the matching or due to restrictive preference cycles, gives rise to a collection of infamously intractable almost-stable matching problems. In practice, however, not every agent is able or likely to initiate deviations caused by blocking pairs. Suppose we knew, for example, due to a set of requirements or estimates based on historical data, which agents are likely to initiate deviations - the deviators - and which are likely to comply with whatever matching they are presented with - the conformists. Can we decide efficiently whether a matching exists in which no deviator is blocking, i.e., in which no deviator has an incentive to initiate a deviation? Furthermore, can we find matchings in which only a few deviators are blocking? We characterise the computational complexity of this question in bipartite and non-bipartite preference settings. Surprisingly, these problems prove computationally intractable in strong ways: for example, unlike in the classical setting, where every agent is considered a deviator, in this extension, we prove that it is NP-complete to decide whether a matching exists where no deviator is blocking. On the positive side, we identify polynomial-time and fixed-parameter tractable cases, providing novel algorithmics for multi-agent systems where stability cannot be fully guaranteed.

翻译：在经典的稳定婚姻与稳定室友问题中，解的规模与稳定性之间存在固有的权衡。虽然在前一问题中，稳定匹配总是存在，且可通过著名的Gale-Shapley算法高效求解；但在后一问题中，稳定匹配的存在性无法保证，但可通过Irving算法高效判定。然而，由于匹配规模存在额外约束或偏好循环的限制，计算最小化不稳定性的匹配会引出一系列众所周知的难解近似稳定匹配问题。然而在实际中，并非所有主体都有能力或可能发起由阻塞对引发的偏离。假设我们基于历史数据的要求或估计，能够识别哪些主体可能发起偏离（即偏离者），哪些可能顺从于给定的匹配（即顺从者）。我们能否高效判定是否存在一个匹配，使得其中没有偏离者构成阻塞，即没有偏离者有动机发起偏离？更进一步，能否找到仅有少数偏离者构成阻塞的匹配？我们在二分与非二分偏好设定下刻画了该问题的计算复杂性。令人惊讶的是，这些问题在计算上表现出强烈的难解性：例如，在经典设定中所有主体均被视为偏离者，而在此扩展模型中，我们证明判定是否存在无偏离者阻塞的匹配是NP完全的。在积极方面，我们识别了多项式时间可解及固定参数可解的情形，为无法完全保证稳定性的多主体系统提供了新的算法思路。