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19
Almost stable matchings by truncating the Gale–Shapley algorithm
 Algorithmica
, 2010
"... We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about ..."
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We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributedsystems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + )approximation algorithm for maximumweight matching in bicoloured graphs and a centralised randomised constanttime approximation scheme for estimating the size of a stable matching. 1
Size Versus Stability in the Marriage Problem
, 2009
"... Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many largescale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maxi ..."
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Cited by 9 (3 self)
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Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many largescale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NPhard and not approximable within n1−ε, for any ε> 0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NPhard and not approximable within δ, for some δ> 1. By contrast, we give a polynomialtime algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimise the number of agents involved in a blocking pair.
Stochastic stability for roommate markets
 Journal of Economic Theory
, 2010
"... Abstract We show that for any roommate market the set of stochastically stable matchings coincides with the set of absorbing matchings. This implies that whenever the core is nonempty (e.g., for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochast ..."
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Abstract We show that for any roommate market the set of stochastically stable matchings coincides with the set of absorbing matchings. This implies that whenever the core is nonempty (e.g., for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochastic stability is a characteristic of the core. Several solution concepts have been proposed to extend the core to all roommate markets (including those with an empty core). An important implication of our results is that the set of absorbing matchings is the only solution concept that is core consistent and shares the stochastic stability characteristic with the core. © 2010 Elsevier Inc. All rights reserved. JEL classification: C62; C71; C78 Keywords: Core; (Pairwise) stability; Roommate markets; Stochastic stability ✩ We thank an associate editor, a referee, Elena Molis (for providing Example 3), Fuhito Kojima, Utku Ünver, and other participants of the SISL miniconference on matching (CalTech) for useful comments and discussions.
Fair Assignment Of Indivisible Objects Under Ordinal Preferences
"... We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of prop ..."
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We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of proportionality and envyfreeness for discrete assignments. The computational complexity of checking whether a fair assignment exists is studied systematically for the fairness notions. We characterize the conditions under which a fair assignment is guaranteed to exist. For a number of fairness concepts, polynomialtime algorithms are presented to check whether a fair assignment exists or not. Our algorithmic results also extend to the case of variable entitlements of agents. Our NPhardness result, which holds for several variants of envyfreeness, answers an open problem posed by
The stable matching problem and its generalizations: an algorithmic and game theoretical approach
, 2007
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The Stability of the Roommate Problem Revisited
, 2007
"... The lack of stability in some matching problems suggests that alternative solution concepts to the core might be applied to find predictable matchings. We propose the absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either g ..."
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Cited by 2 (0 self)
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The lack of stability in some matching problems suggests that alternative solution concepts to the core might be applied to find predictable matchings. We propose the absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts some other matchings when the core is empty. Furthermore, it satisfies an interesting property of outer stability. We also characterize the absorbing sets, determine their number and, in case of multiplicity, we find that they all share a similar structure.
Role Based Hedonic Games
"... This Doctoral Dissertation is brought to you for free and open access by the Computer Science at UKnowledge. It has been accepted for inclusion in Theses and DissertationsComputer Science by an authorized administrator of UKnowledge. For more information, please contact ..."
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This Doctoral Dissertation is brought to you for free and open access by the Computer Science at UKnowledge. It has been accepted for inclusion in Theses and DissertationsComputer Science by an authorized administrator of UKnowledge. For more information, please contact
Solutions for the Stable Roommates Problem with Payments
, 2012
"... KTI/IE Discussion Papers are circulated to promote discussion and provoque comments. Any references to discussion papers should clearly state that the paper is preliminary. Materials published in this series may subject to further publication. ..."
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KTI/IE Discussion Papers are circulated to promote discussion and provoque comments. Any references to discussion papers should clearly state that the paper is preliminary. Materials published in this series may subject to further publication.
Hungarian Academy of Sciences
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