In the stable marriage problem, a set of men and a set of women are given, each of whom has a strictly ordered preference list over the acceptable agents in the opposite class. A matching is called stable if it is not blocked by any pair of agents, who mutually prefer each other to their respective partner. Ties in the preferences allow for three different definitions for a stable matching: weak, strong and super-stability. Besides this, acceptable pairs in the instance can be restricted in their ability of blocking a matching or being part of it, which again generates three categories of restrictions on acceptable pairs. Forced pairs must be in a stable matching, forbidden pairs must not appear in it, and lastly, free pairs cannot block any matching.

Our computational complexity study targets the existence of a stable solution for each of the three stability definitions, in the presence of each of the three types of restricted pairs. We solve all cases that were still open. As a byproduct, we also derive that the maximum size weakly stable matching problem is hard even in very dense graphs, which may be of independent interest.

在稳定的婚姻问题中，给出了一组男人和一组女人，每个人都有一个严格排序的偏好列表，而不是相对阶级中可接受的代理人。如果一个匹配没有被任何一对代理阻止，则称这种匹配为稳定，它们相互之间比彼此更喜欢对方。首选项中的关系允许三种不同的定义来实现稳定的匹配：弱，强和超稳定性。除此之外，实例中的可接受对可以阻止其匹配或成为匹配的一部分的能力受到限制，这又对可接受对产生了三类限制。强制对必须处于稳定的匹配中，禁止对不能出现在其中，最后，自由对不能阻止任何匹配。

我们的计算复杂度研究的目标是，在三种类型的受限对中的每一个都存在的情况下，针对三个稳定性定义中的每一个都存在一个稳定的解决方案。我们解决了所有仍未解决的案件。作为副产品，我们还得出结论，即使在非常稠密的图中，最大尺寸的弱稳定匹配问题也很难解决，而这可能是值得关注的。