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

TL;DR

This paper explores the complexity of approximate solutions for three-dimensional stable matching problems, showing NP-hardness of approximation but also providing polynomial algorithms for constant-factor approximations.

Contribution

It introduces approximation variants of 3GSM and 3PSA, proves their NP-hardness to approximate, and offers simple polynomial algorithms for constant-factor approximations.

Findings

Both maximally stable marriage and matching are NP-hard to approximate within a fixed constant.

A simple polynomial-time algorithm achieves constant-factor approximations for these problems.

The problems are APX-complete, indicating their approximation complexity.

Abstract

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.