Structural aspects of the Student Project Allocation problem

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. Our results extend known structural characterisations from bipartite models to the more complex spa-s setting, and provide a basis for the development of efficient algorithms to address several open problems in spa-s and its extensions.