Meta-rotations and the Structure of Stable Matchings in the Student Project Allocation Problem

We study the Student Project Allocation problem with lecturer preferences over Students (SPA-S), an extension of the well-known Stable Marriage and Hospital Residents problem. In this model, students have preferences over projects, each project is offered by a single lecturer, and lecturers have preferences over students. The goal is to compute a stable matching which is an assignment of students to projects (and thus to lecturers) such that no student or lecturer has an incentive to deviate from their current assignment. While motivated by the university setting, this problem arises in many allocation settings where limited resources are offered by agents with their own preferences, such as in wireless networks. We establish new structural results for the set of stable matchings in SPA-S by developing the theory of meta-rotations, a generalisation of the well-known notion of rotations from the Stable Marriage problem. Each meta-rotation corresponds to a minimal set of changes that transforms one stable matching into another within the lattice of stable matchings. The set of meta-rotations, ordered by their precedence relations, forms the meta-rotation poset. We prove that there is a one-to-one correspondence between the set of stable matchings and the closed subsets of the meta-rotation poset. By developing this structure, we provide a foundation for the design of efficient algorithms for enumerating and counting stable matchings, and for computing other optimal stable matchings, such as egalitarian or minimum-cost matchings, which have not been previously studied in SPA-S.