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 \cite{socialstable} about the existence of a $\frac{3}{2}$-approximation algorithm for the \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：我们针对带有平局的稳定匹配问题（stable matching problem with ties）的多个先前研究的扩展的共同推广形式，给出一个简单的近似算法。这些推广包括图中关键顶点（critical vertices）的存在——我们必须尽可能多地匹配这些顶点，以及不可成为阻塞边的自由边（free edges）和$\Delta$-稳定性（$\Delta$-stabilities），后者意味着一条边要成为阻塞边，其一侧或两侧的改进必须足够大。我们还引入了其他概念以进一步推广这些情形，使得我们的框架能够涵盖众多现有及未来的应用。我们证明，边复制技术（edge duplicating technique）能够使我们同时处理这些不同类型的推广，同时使算法、证明及分析比以往的方法更为简洁。特别地，我们回答了\cite{socialstable}中关于带有自由边的\smti\问题是否存在$\frac{3}{2}$-近似算法的开放问题。这充分表明该技术能很好地把握这些问题的本质，并具有解决未来无数应用的潜力。