Improved approximation results for the stable marriage problem

Halldórsson, Magnús M.; Iwama, Kazuo; Miyazaki, Shuichi; Yanagisawa, Hiroki

doi:10.1145/1273340.1273346

Cited by 62 publications

(56 citation statements)

References 30 publications

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“…For an instance I of SMTI, let opt(I) denote the maximum size of a weakly stable matching in I. Halldorsson et al [6] showed [in the proof of Corollary 3.4] that given an instance I of SMTI of size n, where only one side of the market has agents with indifferences and each of these agents has a single tie of size two, and any arbitrary small positive , it is NP-hard to distinguish between the following two cases: (1) opt(I) ≥ 21− 27 n (2) opt(I) < 19+ 27 n. When choosing so that 0 < < 1 2 we can simplify the above cases to (1) opt(I) > 41 54 n, since opt(I) ≥ 21− 27 n > 41 54 n and (2) opt(I) < 39 54 n, since opt(I) < 19+ 27 n < 39 54 n. Therefore, the number of agents left unmatched on either side of the market is less than 13 54 n in the first case and more than 15 54 n in the second case. Let us now extend instance I to a larger instance of SMTI I as follows.…”

Section: Theoremmentioning

confidence: 99%

“…For the purpose of this paper we assume that the preference lists are complete, i.e., each agent finds each member of the opposite side acceptable. 6 In the stable marriage problem the goal is to compute a stable matching; a matching where no two agents prefer to be matched to each other rather than be matched to their current partners. Unlike most of the literature on stable matching problems [5,10,12], we assume that men and women may have uncertainty in their preferences which can be captured by various probabilistic uncertainty models.…”

Section: Introductionmentioning

confidence: 99%

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Algorithmic Game Theory

Gairing¹,

Savani²

2016

Lecture Notes in Computer Science

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. We consider the two-sided stable matching setting in which there may be uncertainty about the agents' preferences due to limited information or communication. We consider three models of uncertainty:(1) lottery model -in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model -for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model -there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

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Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algorithmic Game Theory

Gairing¹,

Savani²

2016

Lecture Notes in Computer Science

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“…Predictably, however, this is an intractable problem. Irving and Leather [12] have shown that the maximum number of stable matches for the simple marriage matching problem grows exponentially in n (see also [10,11]). Further, they provide a lower bound on the maximum by problem size.…”

Section: Stop Eachmentioning

confidence: 99%

On heuristics for two-sided matching

Kimbrough

Kuo

2010

Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation

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The stable marriage problem is prototypical of two-sided matching problems, widely encountered in practice, in which agents having preferences, interests and capacities for action of their own are paired up or matched. Standardly, variants of the well-known Gale-Shapley deferred acceptance algorithm (GS/DAA) are used to find stable matches. Using evolutionary computation and an agent-based model heuristics, this paper investigates the stable marriage problem as a multiobjective problem, looking at social welfare and equity or fairness, in addition to stability as important aspects of any proposed match. The paper finds that these heuristics are reliably able to discover matches that are Pareto superior to those found by the GS/DAA procedure. Ramifications of this finding are briefly explored, including the question of whether stability in a matching is often strictly required.

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“…When ties are allowed on both sides, the problem is NP-hard to approximate within 33/29 [23] and the currently best known ration is 3/2 [20]. When ties are only allowed on the side of the women the problem is NP-hard to approximate within 21/19 [11] and the currently best known ratio is 25/17 [14].…”

Section: Introductionmentioning

confidence: 99%

Stable Marriage with General Preferences

2016

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Abstract. We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.

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Improved approximation results for the stable marriage problem

Cited by 62 publications

References 30 publications

Algorithmic Game Theory

Algorithmic Game Theory

On heuristics for two-sided matching

Stable Marriage with General Preferences

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