It's Not Easy Being Three: The Approximability of Three-Dimensional Stable Matching Problems

In 1976, Knuth asked if the stable marriage problem (SMP) can be generalized to marriages consisting of 3 genders. In 1988, Alkan showed that the natural generalization of SMP to 3 genders ($3$GSM) need not admit a stable marriage. Three years later, Ng and Hirschberg proved that it is NP-complete to determine if given preferences admit a stable marriage. They further prove an analogous result for the $3$ person stable assignment ($3$PSA) problem. In light of Ng and Hirschberg's NP-hardness result for $3$GSM and $3$PSA, we initiate the study of approximate versions of these problems. In particular, we describe two optimization variants of $3$GSM and $3$PSA: maximally stable marriage/matching (MSM) and maximum stable submarriage/submatching (MSS). We show that both variants are NP-hard to approximate within some fixed constant factor. Conversely, we describe a simple polynomial time algorithm which computes constant factor approximations for the maximally stable marriage and matching problems. Thus both variants of MSM are APX-complete.