Structure and complexity of the student-project allocation problem

Ayegba, Peace (2025) Structure and complexity of the student-project allocation problem. PhD thesis, University of Glasgow.

Abstract

Matching problems occur in many practical settings where agents from one set need to be assigned to agents or resources in another. This thesis presents new results for a class of matching problems known as the Student-Project Allocation problem (spa). In this problem, we are given a set of students, projects, and lecturers, where each project is offered by a single lecturer. Students have preferences over the projects they find acceptable, while lecturers may have no preferences, preferences over students, or preferences over projects. In the spa model where both students and lecturers have preferences, the goal is to find a stable matching, which means an allocation of students to projects such that no student and lecturer would prefer an alternative assignment involving a different project. This thesis explores the complexity and structure of stable matchings in two variants of spa.

In the Student-Project Allocation problem with lecturer preferences over Projects (spa-p), stable matchings may vary in size, and the problem of finding a maximum-size stable matching (denoted max-spa-p) is known to be NP-hard. Another variant is the Student-Project Allocation problem with lecturer preferences over Students, referred to as spa-s. An extension of spa-s where ties are allowed in the preference lists of both students and lecturers is known as spa-st. Similar to the spa-p model, weakly stable matchings in spa-st may differ in size, and it is known that finding a maximum weakly stable matching (denoted max-spa-st) is NP-hard. In both max-spa-p and max-spa-st, we examine how natural restrictions on the preference structure of students and lecturers affect the computational complexity of finding a maximum stable matching. We identify cases that admit polynomial-time algorithms and others that remain NP-hard. In addition, we study the parameterised complexity of max-spa-p, and prove that the problem is fixed-parameter tractable with respect to a natural structural parameter.

Next, we consider the structural aspects of spa-s. It is well known that a single instance may admit multiple stable matchings, and that the number of such matchings may grow exponentially with the input size. We present two new characterisations of the set of stable matchings for any given spa-s instance. First, we prove that the set of stable matchings forms a distributive lattice under a natural dominance relation, in which the student-optimal and lecturer-optimal matchings correspond to the maximum and minimum elements, respectively. In the second characterisation, we extend the notion of rotations, originally defined for the one-to-one Stable Marriage problem, to the more complex spa-s model. We introduce meta-rotations in spa-s, and use this to develop the meta-rotation poset. We prove that there is a one-to-one correspondence between the stable matchings of a given spa-s instance and the closed subsets of the associated meta-rotation poset.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the College of Science and Engineering Scholarship from the University of Glasgow.
• Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
• Colleges/Schools: College of Science and Engineering > School of Computing Science
• Supervisor's Name: Olaosebikan, Dr. Sofiat and Meeks, Professor Kitty
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85664
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 12 Jan 2026 12:10
• Last Modified: 12 Jan 2026 12:15
• Thesis DOI: 10.5525/gla.thesis.85664
• URI: https://theses.gla.ac.uk/id/eprint/85664

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