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The Strongly Stable Roommates Problem

Author Adam Kunysz

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Adam Kunysz

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Adam Kunysz. The Strongly Stable Roommates Problem. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.60

Abstract

An instance of the strongly stable roommates problem with incomplete lists and ties (SRTI) is an undirected non-bipartite graph G = (V,E), with an adjacency list being a linearly ordered list of ties, which are 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} in 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 an O(nm) time algorithm for computing a strongly stable matching, where we denote n = |V| and m = |E|. The best previously known solution had running time O(m^2) [Scott, 2005]. We also give a characterisation of the set of all strongly stable matchings. We show that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and we give an O(nm) algorithm for constructing such a representation. Our algorithms are based on a simple reduction to the bipartite version of the problem.
Keywords
  • strongly stable matching
  • stable roommates
  • rotations
  • matching theory

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