Parameterized algorithms for stable matching with ties and incomplete lists

We study the parameterized complexity of NP-hard optimization versions of Stable Matching and Stable Roommates in the presence of ties and incomplete lists. These problems model many real-life situations where solutions have to satisfy certain predefined criterion of suitability and compatibility. Specifically, our objective is to maximize/minimize the size of the stable matching. Our main theorems state that Stable Matching and Stable Roommates admit small kernels. Consequently, we also conclude that Stable Matching is fixed-parameter tractable (FPT) with respect to solution size, and that Stable Roommates is FPT with respect to a structural parameter. Finally, we analyze the special case where the input graph is planar.