Male Optimal and Unique Stable Marriages with Partially Ordered Preferences

Gelain, Mirco; Pini, Maria Silvia; Rossi, Francesca; Venable, Kristen Brent; Walsh, Toby

doi:10.1007/978-3-642-22427-0_4

Cited by 8 publications

(6 citation statements)

References 12 publications

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“…Manipulation issues have been also considered in the context of stable matching procedures with weighted preferences, where new notions of stability and optimality have been provided [35][36][37]. Besides manipulation, stability, and optimality, also the uniqueness of weakly stable matchings has been studied in the context of stable matching procedures with partially ordered preferences [38,39].…”

Section: Related Workmentioning

confidence: 99%

Local Search Approaches in Stable Matching Problems

et al. 2013

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The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving an SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI (Stable Marriage with Ties and Incomplete lists)) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists, and we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We empirically evaluate our algorithm for SM problems by measuring its runtime behavior and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behavior and its ability to find a maximum cardinality stable marriage. Experimental results suggest that for SM problems, the number of steps of our algorithm grows only as O(n log(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages. Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size, despite the NP-hardness of this problem

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Section: Related Workmentioning

confidence: 99%

Local Search Approaches in Stable Matching Problems

et al. 2013

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“…In SMs, each preference ordering is a strict total order over the members of the other sex. More general notions of SMs allow preference orderings to be partial [41][42][43][44]. This allows for the modeling of both indifference (via ties) and incomparability (via absence of ordering) between members of the other sex.…”

Section: Stable Marriage Problems With Partially Ordered Preferencesmentioning

confidence: 99%

Stability, Optimality and Manipulation in Matching Problems with Weighted Preferences

Pini

Rossi

Venable³

et al. 2013

Algorithms

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The stable matching problem (also known as the stable marriage problem) is a well-known problem of matching men to women, so that no man and woman, who are not married to each other, both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here, we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some real-life situations, it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages that are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that, in most cases, the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. We also consider the manipulability properties of the procedures that return such stable marriages. While we know that all procedures are manipulable by modifying the preference lists or by truncating them, here, we consider if manipulation can occur also by just modifying the weights while preserving the ordering and avoiding truncation. It turns out that, by adding weights, in some cases, we may increase the possibility of manipulating, and this cannot be avoided by any reasonable restriction on the weights

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“…In SMs, each preference ordering is a strict total order over the members of the other sex. More general notions of SMs allow preference orderings to be partial [14,11,10,7,6]. This allows for the modelling of both indifference (via ties) and incomparability (via absence of ordering) between members of the other sex.…”

Section: Stable Marriage Problems With Partially Ordered Preferencesmentioning

confidence: 99%

Stability and Optimality in Matching Problems with Weighted Preferences

Pini

Rossi

Venable

et al. 2013

Communications in Computer and Information Science

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The stable marriage problem is a well-known problem of matching men to women so that no man and woman, who are not married to each other, both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or more generally to any two-sided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some real-life situations it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages which are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that in most cases the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. We also investigate manipulation issues in our framework. More precisely, we adapt the classical notion of manipulation to our context and we study if the procedures which return the new kinds of stable marriages are manipulable.

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Male Optimal and Unique Stable Marriages with Partially Ordered Preferences

Cited by 8 publications

References 12 publications

Local Search Approaches in Stable Matching Problems

Local Search Approaches in Stable Matching Problems

Stability, Optimality and Manipulation in Matching Problems with Weighted Preferences

Stability and Optimality in Matching Problems with Weighted Preferences

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