  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.
