The Strongly Stable Roommates Problem and Linear Programming
Naoyuki Kamiyama
TL;DR
This work tackles the existence of strongly stable matchings in the stable roommates problem with ties. It extends the linear programming framework used for strict preferences in bipartite settings to the non-bipartite case by leveraging the polytope $\mathbf{P}$ and Edmonds-type constraints, enabling a polynomial-time procedure that solves a sequence of LPs to decide existence. The approach builds on self-duality arguments and a decomposition into bipartite components to construct a strongly stable matching when possible. The result provides a new LP-based avenue for stability with ties in non-bipartite graphs, complementing existing combinatorial algorithms and expanding the toolkit for stable matching with ties.
Abstract
The stable roommates problem is a non-bipartite version of the stable matching problem in a bipartite graph. In this paper, we consider the stable roommates problem with ties. In particular, we focus on strong stability, which is one of the main stability concepts in the stable roommates problem with ties. We propose a new polynomial-time algorithm for the problem of checking the existence of a strongly stable matching in the stable roommates problem with ties. More concretely, we extend the linear programming approach of Abeledo and Blum to the stable roommates problem with strict preferences to our problem.