Stable Matching Beyond Bipartite Graphs

Binary matching in bipartite graphs and its extensions have been well studied over the decades. A stable matching (or marriage) seeks to establish a stable binary pairing of two genders, where each member in a gender has a preference list for the other gender. In addition, all members are paired (also called perfect matching), with one selection made from each gender. Gale and Shapley showed that a stable marriage exists for any set of preferences in any complete and balanced bipartite graphs. However, stable matching may not exist in a non-bipartite graph represented by a complete graph (i.e. single gender), as shown in the stable roommates problem. In this paper, we study binary matching in complete and balanced k-partite graphs with an even number of members, and show that except when k = 2, preference lists always exist under which a stable binary matching does not exist. We use a solution for the stable roommates problem to illustrate the matching identification process if one exists. Under a natural extension of pairwise stability for binary matching to k-ary matching, we show that stable k-ary matching exists for any preference lists in any complete and balanced k-partite graphs. Here, we assume that there is a strict preference order of the members over all individual members from different genders, as opposed to preference order over a combination of members from different genders used in existing multi-dimensional extensions. An extended Gale-Shapley algorithm is introduced, together with various implementations, to determine such a stable k-ary matching. A parallel implementation is also proposed to speed up the matching process. In the extension, we further propose a weakened unstable condition for k-ary matching and show the existence of stable k-ary matching.