Kwanashie, Augustine
(2015)
*Efficient algorithms for optimal matching problems under preferences.*
PhD thesis, University of Glasgow.

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## Abstract

In this thesis we consider efficient algorithms for matching problems involving preferences,

i.e., problems where agents may be required to list other agents that they find

acceptable in order of preference. In particular we mainly study the Stable Marriage

problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation

problem (SPA), and some of their variants. In some of these problems the aim

is to find a stable matching which is one that admits no blocking pair. A blocking pair

with respect to a matching is a pair of agents that prefer to be matched to each other

than their assigned partners in the matching if any.

We present an Integer Programming (IP) model for the Hospitals / Residents problem

with Ties (HRT) and use it to find a maximum cardinality stable matching. We also

present results from an empirical evaluation of our model which show it to be scalable

with respect to real-world HRT instance sizes.

Motivated by the observation that not all blocking pairs that exist in theory will lead

to a matching being undermined in practice, we investigate a relaxed stability criterion

called social stability where only pairs of agents with a social relationship have the

ability to undermine a matching. This stability concept is studied in instances of

the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We

show that, in the smi and hr contexts, socially stable matchings can be of varying

sizes and the problem of finding a maximum socially stable matching (max smiss and

max hrss respectively) is NP-hard though approximable within 3/2. Furthermore we

give polynomial time algorithms for three special cases of the problem arising from

restrictions on the social network graph and the lengths of agents’ preference lists.

We also consider other optimality criteria with respect to social stability and establish

inapproximability bounds for the problems of finding an egalitarian, minimum regret

and sex equal socially stable matching in the sm context.

We extend our study of social stability by considering other variants and restrictions

of max smiss and max hrss. We present NP-hardness results for max smiss even

under certain restrictions on the degree and structure of the social network graph as

well as the presence of master lists. Other NP-hardness results presented relate to the

problem of determining whether a given man-woman pair belongs to a socially stable

matching and the problem of determining whether a given man (or woman) is part of

at least one socially stable matching. We also consider the Stable Roommates problem

with Incomplete lists under Social Stability (a non-bipartite generalisation of smi under

social stability). We observe that the problem of finding a maximum socially stable

matching in this context is also NP-hard. We present efficient algorithms for three

special cases of the problem arising from restrictions on the social network graph and

the lengths of agents’ preference lists. These are the cases where (i) there exists a

constant number of acquainted pairs (ii) or a constant number of unacquainted pairs

or (iii) each preference list is of length at most 2.

We also present algorithmic results for finding matchings in the spa context that are

optimal with respect to profile, which is the vector whose ith component is the number

of students assigned to their ith-choice project. We present an efficient algorithm for

finding a greedy maximum matching in the spa context — this is a maximum matching

whose profile is lexicographically maximum. We then show how to adapt this algorithm

to find a generous maximum matching — this is a matching whose reverse profile is

lexicographically minimum. We demonstrate how this approach can allow additional

constraints, such as lecturer lower quotas, to be handled flexibly. We also present

results of empirical evaluations carried out on both real world and randomly generated

datasets. These results demonstrate the scalability of our algorithms as well as some

interesting properties of these profile-based optimality criteria.

Practical applications of spa motivate the investigation of certain special cases of the

problem. For instance, it is often desired that the workload on lecturers is evenly distributed

(i.e. load balanced). We enforce this by either adding lower quota constraints

on the lecturers (which leads to the potential for infeasible problem instances) or adding

a load balancing optimisation criterion. We present efficient algorithms in both cases.

Another consideration is the fact that certain projects may require a minimum number

of students to become viable. This can be handled by enforcing lower quota constraints

on the projects (which also leads to the possibility of infeasible problem instances). A

technique of handling this infeasibility is the idea of closing projects that do not meet

their lower quotas (i.e. leaving such project completely unassigned). We show that the

problem of finding a maximum matching subject to project lower quotas where projects

can be closed is NP-hard even under severe restrictions on preference lists lengths and

project upper and lower quotas. To offset this hardness, we present polynomial time

heuristics that find large feasible matchings in practice. We also present ip models

for the spa variants discussed and show results obtained from an empirical evaluation

carried out on both real and randomly generated datasets. These results show that

our algorithms and heuristics are scalable and provide good matchings with respect to

profile-based optimality

Item Type: | Thesis (PhD) |
---|---|

Qualification Level: | Doctoral |

Keywords: | stable matching, student project allocation, hospitals residents problem, algorithms, matching |

Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Computer software |

Colleges/Schools: | College of Science and Engineering > School of Computing Science |

Supervisor's Name: | Manlove, Dr. David |

Date of Award: | 2015 |

Depositing User: | Mr Augustine Kwanashie |

Unique ID: | glathesis:2015-6706 |

Copyright: | Copyright of this thesis is held by the author. |

Date Deposited: | 25 Sep 2015 07:25 |

Last Modified: | 05 Oct 2015 11:45 |

URI: | https://theses.gla.ac.uk/id/eprint/6706 |

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