# The stable marriage problem with ties and restricted edges

## TL;DR

This paper investigates the computational complexity of finding stable matchings in the stable marriage problem with ties and restricted pairs, solving previously open cases and analyzing the problem's difficulty in dense graphs.

## Contribution

It provides a comprehensive complexity analysis for all stability and restriction combinations, solving open problems and highlighting the hardness in dense graphs.

## Key findings

- All open cases of the problem are solved.
- Maximum size weakly stable matching is NP-hard in dense graphs.
- The study clarifies the complexity landscape of stable matchings with restrictions.

## Abstract

In the stable marriage problem, a set of men and a set of women are given, each of whom has a strictly ordered preference list over the acceptable agents in the opposite class. A matching is called stable if it is not blocked by any pair of agents, who mutually prefer each other to their respective partner. Ties in the preferences allow for three different definitions for a stable matching: weak, strong and super-stability. Besides this, acceptable pairs in the instance can be restricted in their ability of blocking a matching or being part of it, which again generates three categories of restrictions on acceptable pairs. Forced pairs must be in a stable matching, forbidden pairs must not appear in it, and lastly, free pairs cannot block any matching. Our computational complexity study targets the existence of a stable solution for each of the three stability definitions, in the presence of each of the three types of restricted pairs. We solve all cases that were still open. As a byproduct, we also derive that the maximum size weakly stable matching problem is hard even in very dense graphs, which may be of independent interest.