A necessary and sufficient condition for the existence of a complete stable matching

Abstract

The stable roommates problem is a well-known problem of matching n people into n 2 disjoint pairs so that no two unmatched persons both prefer each other to their partners under the matching. We call such a matching “ a complete stable matching.” It is known that a complete stable matching may not exist. Irving described an O( n 2) algorithm that would find one complete stable matching if there is one, or would report that none exists. In this paper, we give a necessary and sufficient condition for the existence of a complete stable matching; namely, the non-existence of any odd party, which will be defined subsequently. We define a new structure called a “stable partition,” which generalizes the notion of a complete stable matching, and prove that every instance of the stable roommates problem has at least one such structure. We also show that a stable partition contains all the odd parties, if there are any. Finally we have an O( n 2) algorithm that finds one stable partition which in turn gives all the odd parties.