A necessary and sufficient condition for the existence of a complete stable matching

doi:10.1016/0196-6774(91)90028-W

Journal of Algorithms

Volume 12, Issue 1, March 1991, Pages 154-178

https://doi.org/10.1016/0196-6774(91)90028-W Get rights and content

Abstract

The stable roommates problem is a well-known problem of matching n people into $n 2$ disjoint pairs so that no two unmatched persons both prefer each other to their partners under the matching. We call such a matching “a complete stable matching.” It is known that a complete stable matching may not exist. Irving described an O(n²) algorithm that would find one complete stable matching if there is one, or would report that none exists. In this paper, we give a necessary and sufficient condition for the existence of a complete stable matching; namely, the non-existence of any odd party, which will be defined subsequently. We define a new structure called a “stable partition,” which generalizes the notion of a complete stable matching, and prove that every instance of the stable roommates problem has at least one such structure. We also show that a stable partition contains all the odd parties, if there are any. Finally we have an O(n²) algorithm that finds one stable partition which in turn gives all the odd parties.

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This paper studies a multilateral matching market in which each participant can sign contracts with any other agents. This market subsumes the two-sided matching and the roommate problem as special cases. We consider a hypergraph $(I, C)$ , with possible multiple (hyper)edges and loops, in which the vertices $i \in I$ are interpreted as agents, and the edges $c \in C$ as contracts that can be concluded between agents. The preferences of each agent $i$ are given by a choice function $f_{i}$ possessing the so-called path independent property. In this general setup we consider the notion of stable contract network.
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Citation Excerpt :
A classical real-world example is the problem of finding roommates at the university, hence the name of the problem [4]. In contrast to the SMP, the RP have instances where a stable solution may fail to exist [5]. In addition to the stable solutions, in both the bipartite and monopartite models, it is possible to study the ground state solution, that is the configuration that maximizes the global benefit of the society.
In the matching problem agents with partially overlapping interests must be matched pairwise. It has inspired many physicists working on complex systems who studied the properties of the stable state and the ground state by employing the tools of statistical mechanics. Here, we examine the matching problem from a different perspective by studying a dynamic evolution of a matching system. We propose a model where agents interact locally and selfishly to maximize their benefit. We investigate the dynamic and steady-state properties of our model in two different cases: when mutual benefits between agents are symmetrical and when they are not. In particular, we show analytically that the global benefit of the society in the stationary state is far from the ground state in both cases, and this distance increases with the number of agents. However, a society with symmetrical interests performs better than one with asymmetrical interests. Possible practical implications of our findings are discussed.
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Appendices A–C provide the proofs. Appendix D discusses the conditions for Tan's (1991) results, which we heavily exploit in our analysis. In the literature, a number of studies have defined stability concepts based on chains of deviations and their final outcomes, in a similar spirit with ours.
We propose a new solution concept in the roommate problem, based on the “robustness” of deviations (i.e., blocking coalitions). We call a deviation from a matching robust up to depth k, if none of the deviators gets worse off than at the original matching after any sequence of at most k subsequent deviations. We say that a matching is stable against robust deviations (for short, SaRD) up to depth k, if no deviation from it is robust up to depth k. As a smaller k imposes a stronger requirement for a matching to be SaRD, we investigate the existence of a matching that is SaRD with a minimal depth k. We constructively demonstrate that a SaRD matching always exists for $k = 3$ and establish sufficient conditions for $k = 1$ and 2.

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^∗: This research was supported by National Science Council of Republic of China under Grant NSC 77-0408-E009-18.

View full text

A necessary and sufficient condition for the existence of a complete stable matching

Abstract

Discrete Appl. Math.

J. Algorithms

College admissions and the stability of marriage

Amer. Math. Monthly

Three fast algorithms for four problems in stable marriage

SIAM J. Comput.

The structure of the stable roommates problem: Efficient representation and enumeration of all stable assignments

SIAM J. Comput.

The Stable Marriage Problem: Structure and Algorithms