Structural aspects of the Student Project Allocation problem

Abstract

We study the Student Project Allocation problem with lecturer preferences over Students (SPA-S), which involves the assignment of students to projects based on student preferences over projects, lecturer preferences over students, and capacity constraints on both projects and lecturers. The goal is to find a stable matching that ensures no student and lecturer can mutually benefit by deviating from a given assignment to form an alternative arrangement involving some project. We explore the structural properties of SPA-S and characterise the set of stable matchings for an arbitrary SPA-S instance. We prove that, similar to the classical Stable Marriage problem (SM) and the Hospital Residents problem (HR), the set of all stable matchings in SPA-S forms a distributive lattice. In this lattice, the student-optimal and lecturer-optimal stable matchings represent the minimum and maximum elements, respectively. Finally, we introduce meta-rotations in the SPA-S setting using illustrations, demonstrating how they capture the relationships between stable matchings. These novel structural insights paves the way for efficient algorithms that address several open problems related to stable matchings in SPA-S.