# Characterisation of Strongly Stable Matchings

@inproceedings{Kunysz2015CharacterisationOS,
title={Characterisation of Strongly Stable Matchings},
author={Adam Kunysz and Katarzyna E. Paluch and Pratik Ghosal},
booktitle={ACM-SIAM Symposium on Discrete Algorithms},
year={2015}
}
• Published in
ACM-SIAM Symposium on…
1 June 2015
• Mathematics
An instance of a strongly stable matching problem (SSMP) is an undirected bipartite graph G = (A∪B, E), with an adjacency list of each vertex being a linearly ordered list of ties, which are subsets of vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge (x, y) ∈ E\M is a blocking edge for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly…
12 Citations
• Mathematics
• 2021
A polyhedral characterisation of the set of allsuper-stable matchings is given and it is proved that the super-stable matching polytope is integral, thus solving an open problem stated in the book by Gusfield and Irving.
It is shown that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and an O(nm) algorithm for constructing such a representation is given.
The main result of this paper is an efficient O(nm log (Wn) time algorithm for computing a maximum weight strongly stable matching, where n = |V |, m = |E| and W is amaximum weight of an edge in G, which shows that the problem can be solved in O (nm) time.
The notion of strong stability is investigated and an algorithm for computing a strongly stable b-matching optimal for vertices of A is given, where m = |E|, which improves on the previous algorithm by Chen and Ghosh.
This work considers a many-to-one variant of the stable matching problem where one side has a matroid constraint, and proposes a polynomial-time algorithm for this problem.
• Mathematics, Computer Science
ArXiv
• 2022
This work employs the theory of rotations for Stable Roommates to develop a polynomial-time algorithm for adapting StableRoommates matchings to forced pairs and shows that the same problem for forbidden pairs is NP-hard.
• Economics
MFCS
• 2022
When computing stable matchings, it is usually assumed that the preferences of the agents in the matching market are fixed. However, in many realistic scenarios, preferences change over time.
• Mathematics
STACS
• 2019
By designing three polynomial algorithms and two NP-completeness proofs, this work determines the complexity of all cases not yet known, and thus gives an exact boundary in terms of preference structure between tractable and intractable cases.
• Economics
2019 10th International Conference on Computing, Communication and Networking Technologies (ICCCNT)
• 2019
An algorithm is proposed to rank the choices of a tenant or house owner by the preferences they desire, used here for the matching between the tenants and house owners so that there will be no unstable tenant-owner pair.
• Economics
Oper. Res.
• 2022
This work investigates two extensions introduced in this framework -- legal assignments and the EADAM algorithm -- through the lens of classical theory of stable matchings, and proves that the set ${\cal L}$ is exactly the set of stable assignments in another instance.

## References

SHOWING 1-10 OF 24 REFERENCES

• Mathematics
IPCO
• 2014
We consider the problem of computing a large stable matching in a bipartite graph G = (A ∪ B, E) where each vertex u ∈ A ∪ B ranks its neighbors in an order of preference, perhaps involving ties. A
An instance of the classical stable marriage problem involves n men and n women, and each person ranks all n members of the opposite sex in strict order of preference. The effect of allowing ties in
• Computer Science
TALG
• 2007
An algorithm is given for computing strongly stable matchings in the hospitals-residents setting, where n is the number of vertices and m the Number of edges, where there is no blocking edge with respect to it.
• Computer Science
ICALP
• 1999
This paper shows that the situation changes substantially if the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N1-Ɛ, where N is the instance size, unless P=NP.
A two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching, is considered, and a very natural, economically reasonable, local, linear time algorithm is presented.
A 3/2-approximation algorithm for stable matchings that runs in O(m) time is given and the extension of the algorithm for computing stable many-to-many matchings is given.
• Computer Science, Mathematics
• 2003
A new algorithm is presented to maintain the topological order of a directed acyclic graph (DAG) in the presence of edge insertions and deletions and a empirical comparison against three existing solutions using random DAG’s is provided.