The Meta-rotation Poset for Student-Project Allocation
Peace Ayegba, Sofiat Olaosebikan
TL;DR
This paper addresses stable matchings in the spa-s setting, where each project is offered by a lecturer and both students and lecturers have preferences, by introducing meta-rotations as a spa-s-specific generalisation of rotations. It defines an exposed meta-rotation and proves that eliminating such a meta-rotation from a stable matching yields another stable matching that is weakly worse for all students, with the original matching dominating the new one. The authors establish a meta-rotation poset $\Pi(I)$ in which each stable matching corresponds to a closed subset, enabling enumeration of all stable matchings and construction of optimal matchings (e.g., egalitarian or minimum-cost) via controlled elimination sequences. The framework overcomes structural differences between spa-s and its predecessors (sm/hr) by carefully accounting for project and lecturer capacities, undersubscription effects, and the notion of next possible assignments. These insights open avenues for polyhedral formulations and efficient algorithms for counting and optimizing across all stable spa-s matchings with potential practical impact in resource allocation settings like education and networks.
Abstract
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.