We consider the stable marriage problem in the presence of ties in preferences and critical vertices. The input to our problem is a bipartite graph G = (A U B, E) where A and B denote sets of vertices which need to be matched. Each vertex has a preference ordering over its neighbours possibly containing ties. In addition, a subset of vertices in A U B are marked as critical and the goal is to output a matching that matches as many critical vertices as possible. Such matchings are called critical matchings in the literature and in our setting, we seek to compute a matching that is critical as well as optimal with respect to the preferences of the vertices. Stability, which is a well-accepted notion of optimality in the presence of two-sided preferences, is generalized to weak-stability in the presence of ties. It is well known that in the presence of critical vertices, a matching that is critical as well as weakly stable may not exist. Popularity is another well-investigated notion of optimality for the two-sided preference list setting, however, in the presence of ties (even with no critical vertices), a popular matching need not exist. We, therefore, consider the notion of relaxed stability which was introduced and studied by Krishnaa et. al. (SAGT 2020). We show that a critical matching which is relaxed stable always exists in our setting although computing a maximum-sized relaxed stable matching turns out to be NP-hard. Our main contribution is a 3/2 approximation to the maximum-sized critical relaxed stable matching for the stable marriage problem with two-sided ties and critical vertices.

翻译：我们研究存在平局偏好与关键顶点的稳定婚姻问题。问题输入为二部图G = (A U B, E)，其中A和B表示需要匹配的顶点集合。每个顶点对其邻居存在可能包含平局的偏好序。此外，A U B中的一部分顶点被标记为关键顶点，目标是输出尽可能匹配更多关键顶点的匹配。此类匹配在文献中被称为关键匹配，在我们的设定中，需要计算同时满足关键性与偏好最优性的匹配。稳定性作为双侧偏好场景中公认的最优性概念，在存在平局时被推广为弱稳定性。众所周知，当存在关键顶点时，同时满足关键性与弱稳定性的匹配可能不存在。流行性作为双侧偏好列表设定中另一个被充分研究的最优性概念，在存在平局时（即使没有关键顶点）也可能不存在流行匹配。因此，我们考虑由Krishnaa等人（SAGT 2020）引入并研究的松弛稳定性概念。我们证明在该设定中始终存在满足松弛稳定性的关键匹配，尽管计算最大基数松弛稳定匹配被证明是NP难问题。我们的主要贡献是为具有双侧平局与关键顶点的稳定婚姻问题，提出了一个达到最大基数关键松弛稳定匹配3/2近似比的算法。