A probabilistic approach to the stable matching problem has been identified as an important research area with several important open problems. When considering random matchings, ex-post stability is a fundamental stability concept. A prominent open problem is characterizing ex-post stability and establishing its computational complexity. We investigate the computational complexity of testing ex-post stability. Our central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete. The result even holds if both sides have dichotomous preferences. On the positive side, we give an algorithm using an integer programming approach, that can determine a decomposition with a maximum probability of being weakly stable. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.

翻译：稳定匹配问题的概率化方法已被视为一个重要的研究领域，其中存在若干关键未解难题。在考虑随机匹配时，事后稳定性是一个基础性的稳定概念。当前一个突出的开放性问题在于刻画事后稳定性的特征并确定其计算复杂性。本文研究了检验事后稳定性的计算复杂性。我们的核心结果表明：当任意一方在偏好/优先级中存在并列关系时，检验事后稳定性是NP完全问题。该结论甚至在双方均具有二分偏好的情况下依然成立。在积极方面，我们提出了一种基于整数规划方法的算法，能够确定具有最大弱稳定概率的分解方案。我们还考虑了更强版本的事后稳定性（特别是鲁棒事后稳定性与事后强稳定性），并证明它们可在多项式时间内完成检验。