The complexity of counting stable marriages
SIAM Journal on Computing
The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
A new fixed point approach for stable networks stable marriages
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Median graphs, parallelism and posets
Discrete Mathematics
A new fixed point approach for stable networks and stable marriages
Journal of Computer and System Sciences
A New Approach to Stable Matching Problems
SIAM Journal on Computing
The Geometry of Fractional Stable Matchings and its Applications
Mathematics of Operations Research
Stable Networks and Product Graphs
Stable Networks and Product Graphs
The Art of Computer Programming, Volume 4, Fascicle 0: Introduction to Combinatorial Algorithms and Boolean Functions (Art of Computer Programming)
The generalized median stable matchings: finding them is not that easy
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Understanding the Generalized Median Stable Matchings
Algorithmica - Special Issue: Matching Under Preferences; Guest Editors: David F. Manlove, Robert W. Irving and Kazuo Iwama
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For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are stable matchings that assign each participant to his or her (lower/upper) median stable partner. Moreover, for SM instances, a stable matching has this property if and only if it is a median of the distributive lattice formed by the instance's stable matchings. In this paper, we show that the above local/global median phenomenon first observed in SM stable matchings also extends to SR stable matchings because SR stable matchings form a median graph. In the course of our investigations, we also prove that three seemingly different structures are pairwise duals of each other—median graphs give rise to mirror posets and vice versa, and mirror posets give rise to SR stable matchings and vice versa. Together, they imply that for every median graph $G$ with $n$ vertices, there is an SR instance $I(G)$ with $O(n^2)$ participants whose graph of stable matchings is isomorphic to $G$. Our results are analogous to the pairwise duality results known for distributive lattices, posets, and SM stable matchings. Interestingly, some of these results can also be inferred from the work of Feder in the early 1990s. Our constructions and proofs, however, are more natural generalizations of those used for SM instances.