Stable Marriage with Covering Constraints–A Complete Computational Trichotomy

Mnich, Matthias; Schlotter, Ildikó

doi:10.1007/978-3-319-66700-3_25

Cited by 14 publications

(7 citation statements)

References 41 publications

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“…Very recently, five different works have studied hard stable matching problems from the perspective of parameterised complexity. In [29], the authors obtained results on the parameterised complexity of finding a stable matching which matches a given set of distinguished agents and has as few blocking pairs as possible. In [30] it is shown that several hard stable matching problems, including MAX SMTI, are W[1]-hard when parameterised by the treewidth of the graph obtained by adding an edge between each pair of agents that find each other mutually acceptable.…”

Section: Related Workmentioning

confidence: 99%

Solving hard stable matching problems involving groups of similar agents

Meeks

Rastegari

2020

Theoretical Computer Science

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Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents' attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), several well-studied NP-hard stable matching problems including MAX SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belong to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. For MAX SMTI this tractability result can be extended to the setting in which each agent promotes at most one "exceptional" candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NPhard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists, even if the number of types is bounded by a constant.

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Section: Related Workmentioning

confidence: 99%

Solving hard stable matching problems involving groups of similar agents

Meeks

Rastegari

2020

Theoretical Computer Science

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“…Very recently, three different works have studied hard stable matching problems from the perspective of parameterised complexity. Mnich and Schlotter [19] obtained results on the parameterised complexity of finding a stable matching which matches a given set of distinguished agents and has as few blocking pairs as possible. Gupta et al [13] showed that several hard stable matching problems, including Max SMTI, are W[1]-hard when parameterised by the treewidth of the graph obtained by adding an edge between each pair of agents that find each other mutually acceptable.…”

Section: Related Workmentioning

confidence: 99%

Stable Marriage with Groups of Similar Agents

Meeks

Rastegari

2018

Web and Internet Economics

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Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents' attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), the well-known NP-hard matching problem Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belongs to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. This tractability result can be extended to the setting in which each agent promotes at most one "exceptional" candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists.

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“…By adapting our reduction, we also answer Our results are summarized in Table 1. Besides the relevant work mentioned above there is a growing body of research regarding the parameterized complexity of preference-based stable matching problems [42,43,46,45,26,15].…”

Section: Our Contributionsmentioning

confidence: 99%

How hard is it to satisfy (almost) all roommates?

Chen,

Hermelin,

Sorge

et al. 2017

Preprint

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The classic Stable Roommates problem (which is the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners.Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egalitarian Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost γ, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable Roommates, which seeks to find a matching with a minimum number β of blocking pairs. Our main result is that Egalitarian Stable Roommates parameterized by γ is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by β is W[1]-hard, even if the length of each preference list is at most five.

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Stable Marriage with Covering Constraints–A Complete Computational Trichotomy

Cited by 14 publications

References 41 publications

Solving hard stable matching problems involving groups of similar agents

Solving hard stable matching problems involving groups of similar agents

Stable Marriage with Groups of Similar Agents

How hard is it to satisfy (almost) all roommates?

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