We give a simple approximation algorithm for a common generalization of many previously studied extensions of the maximum size stable matching problem with ties. These generalizations include the existence of critical vertices in the graph, amongst whom we must match as much as possible, free edges, that cannot be blocking edges and $\Delta$-stabilities, which mean that for an edge to block, the improvement should be large enough on one or both sides. We also introduce other notions to generalize these even further, which allows our framework to capture many existing and future applications. We show that the edge duplicating technique allows us to treat these different types of generalizations simultaneously, while also making the algorithm, the proofs and the analysis much simpler and shorter than in previous approaches. In particular, we answer an open question by Askalidis et al. (2013) about the existence of a $\frac{3}{2}$-approximation algorithm for the MAX-SMTI problem with free edges. This demonstrates that this technique can grasp the underlying essence of these problems quite well and have the potential to be able to solve many future applications.

翻译：我们提出了一种简单近似算法，用于解决最大规模带平局稳定匹配问题中许多先前研究的扩展问题的共同推广形式。这些推广包括图中存在关键顶点（我们必须尽可能在这些顶点之间进行匹配）、自由边（不能作为阻塞边）以及Δ稳定性（即一条边要成为阻塞边，其改进幅度必须在一侧或两侧足够大）。我们还引入了其他概念进一步推广这些性质，使我们的框架能够涵盖现有和未来的许多应用场景。我们证明，边复制技术能够同时处理这些不同类型的推广，同时使算法、证明及分析过程较先前方法更为简洁高效。特别地，我们解决了Askalidis等人（2013年）关于存在自由边的MAX-SMTI问题是否具有3/2近似算法的开放性问题。这表明该技术能够很好地把握这些问题的本质，并具备解决未来诸多应用问题的潜力。