The center stable matchings and the centers of cover graphs of distributive lattices

Let I be an instance of the stable marriage (SM) problem. In the late 1990s, Teo and Sethuraman discovered the existence of median stable matchings, which are stable matchings that match all participants to their (lower/upper) median stable partner. About a decade later, Cheng showed that not only are they locally-fair, but they are also globally-fair in the following sense: when G(I) is the cover graph of the distributive lattice of stable matchings, these stable matchings are also medians of G(I) – i.e., their average distance to the other stable matchings is as small as possible. Unfortunately, finding a median stable matching of I is #P-hard. Inspired by the fairness properties of the median stable matchings, we study the center stable matchings which are the centers of G(I) – i.e., the stable matchings whose maximum distance to any stable matching is as small as possible. Here are our two main results. First, we show that a center stable matching of I can be computed in O(|I|) time. Thus, center stable matchings are the first type of globally-fair stable matchings we know of that can be computed efficiently. Second, we show that in spite of the first result, there are similarities between the set of median stable matchings and the set of center stable matchings of I. The former induces a hypercube in G(I) while the latter is the union of hypercubes of a fixed dimension in G(I). Furthermore, center stable matchings have a property that approximates the one found by Teo and Sethuraman for median stable matchings. Finally, we note that our results extend to other variants of SM whose solutions form a distributive lattice and whose rotation posets can be constructed efficiently.