We give a simple approximation algorithm for a common generalization of many previously studied extensions of the 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 our 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 then 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 well that this technique can grasp the underlying essence of these problems quite well and have the potential to be able to solve countless future applications as well.

翻译：我们提出了一种简单近似算法，用于统一处理带选项稳定匹配问题中许多此前研究过的扩展形式。这些扩展包括：图中存在关键顶点（需尽可能匹配更多这类顶点）、自由边（不能成为阻塞边）以及Δ-稳定性（即要求边成为阻塞边时，对一侧或两侧的改进幅度需足够大）。我们还引入其他概念进一步推广这些扩展，使我们的框架能够涵盖现有及未来的诸多应用场景。研究表明，我们提出的边复制技术可同时处理这些不同类型的扩展问题，同时使算法、证明与分析相较于此前方法更加简洁。特别地，我们解答了Askalidis等人（2013年）提出的关于含自由边的Max-SMTI问题是否存在3/2近似算法的公开问题。这充分表明该技术能精准把握此类问题的内在本质，并具备解决未来大量潜在应用问题的能力。