Three-Dimensional Stable Matching Problems

The stable marriage problem is a matching problem that pairs members of two sets. The objective is to achieve a matching that satis es all participants based on their preferences. The stable roommate problem is a variant involving only one set, which is partitioned into pairs with a similar objective. There exist asymptotically optimal algorithms that solve both problems. In this paper, we investigate the complexity of three-dimensional extensions of these problems. This is one of twelve research directions suggested by Knuth in his book on the stable marriage problem. We show that these problems are NP-complete, and hence it is unlikely that there exist e cient algorithms for their solutions. The approach developed in this paper provides an alternateNP-completeness proof for the hospitals/residents problem with couples|an important practical problem shown earlier to be NP-complete by E. Ronn.