• DOI:10.1137/1.9781611974331.CH8
• Corpus ID: 314539

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}
}

• Adam Kunysz, Katarzyna E. Paluch, Pratik Ghosal
• 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 prefers x to its current partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We present a characterisation of the set of all strongly stable matchings, thus solving an open problem already stated in the book by Gusfield and Irving [7]. It has previously been shown that strongly stable matchings form a distributive lattice [8] and although the number of strongly stable matchings can be exponential in the number of vertices, we show that there exists a partial order with O(m) elements representing all strongly stable matchings, where m denotes the number of edges in the graph. We give two algorithms that construct two such representations: one in O(nm2) time and the other in O(nm) time, where n denotes the number of vertices in the graph. Note that the construction of the second representation has the same time complexity as that of computing a single strongly stable matching.