For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are stable matchings that assign each participant to his or her (lower/upper) median stable partner. Moreover, for SM instances, a stable matching has this property if and only if it is a median of the distributive lattice formed by the instance’s stable matchings. In this paper, we show that the above local/global median phenomenon first observed in SM stable matchings also extends to SR stable matchings because SR stable matchings form a median graph. In the course of our investigations, we also prove that three seemingly different structures are pairwise duals of each other – median graphs give rise to mirror posets and vice versa, and mirror posets give rise to SR stable matchings and vice versa. Together, they imply that for every median graph G with n vertices, there is an SR instance I(G) with O(n2 ) participants whose graph of stable matchings is isomorphic to G. Our results are...
Christine T. Cheng, Anhua Lin