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Solving hard stable matching problems involving groups of similar agents

Solving hard stable matching problems involving groups of similar agents
Solving hard stable matching problems involving groups of similar agents
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 NP-hard 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.
Fixed-parameter tractable algorithms, NP-hard problems, Parameterized complexity, Stable marriage problem, Stable matchings, Stable roommate problem
0304-3975
171-194
Meeks, Kitty
1ee19992-11dc-44ba-b8d0-c275f2009448
Rastegari, Baharak
6ba9e93c-53ba-4090-8f77-c1cb1568d7d1
Meeks, Kitty
1ee19992-11dc-44ba-b8d0-c275f2009448
Rastegari, Baharak
6ba9e93c-53ba-4090-8f77-c1cb1568d7d1

Meeks, Kitty and Rastegari, Baharak (2020) Solving hard stable matching problems involving groups of similar agents. Theoretical Computer Science, 844, 171-194. (doi:10.1016/j.tcs.2020.08.017).

Record type: Article

Abstract

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 NP-hard 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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Accepted/In Press date: 17 August 2020
e-pub ahead of print date: 19 August 2020
Published date: 6 December 2020
Keywords: Fixed-parameter tractable algorithms, NP-hard problems, Parameterized complexity, Stable marriage problem, Stable matchings, Stable roommate problem

Identifiers

Local EPrints ID: 444481
URI: http://eprints.soton.ac.uk/id/eprint/444481
ISSN: 0304-3975
PURE UUID: 5a5e9492-8f57-45fc-b642-09423fe830a7
ORCID for Baharak Rastegari: ORCID iD orcid.org/0000-0002-0985-573X

Catalogue record

Date deposited: 21 Oct 2020 16:30
Last modified: 26 Nov 2021 05:43

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Contributors

Author: Kitty Meeks
Author: Baharak Rastegari ORCID iD

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