Extending Stable and Popular Matching Algorithms from Bipartite to Arbitrary Instances

TL;DR

This paper extends key algorithms for stable and popular matchings from bipartite graphs to arbitrary graphs, solving longstanding open problems and improving matching solutions in complex scenarios like roommate problems with ties.

Contribution

It introduces extensions of the 3/2-approximation algorithm and popular matching algorithms from bipartite to arbitrary graphs, addressing open questions in the field.

Findings

Extended the 3/2-approximation algorithm to roommates with ties

Adapted popular matching algorithms to arbitrary graphs

Solved a 20-year-old open problem in stable matchings

Abstract

We consider stable and popular matching problems in arbitrary graphs, which are referred to as stable roommates instances. We extend the 3/2-approximation algorithm for the maximum size weakly stable matching problem to the roommates case, which solves a more than 20 year old open question of Irving and Manlove about the approximability of maximum size weakly stable matchings in roommates instances with ties [Irving and Manlove 2002] and has nice applications for the problem of matching residents to hospitals in the presence of couples. We also extend the algorithm that finds a maximum size popular matching in bipartite graphs in the case of strict preferences and the algorithm to find a popular matching among maximum weight matchings. While previous attempts to extend the idea of promoting the agents or duplicating the edges from bipartite instances to arbitrary ones failed, these results show that with the help of a simple observation, we can indeed bridge the gap and extend these algorithms