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.

翻译：本文研究带导师对学生偏好的学生项目分配问题（SPA-S），该问题涉及根据学生对项目的偏好、导师对学生的偏好以及项目和导师的容量约束，将学生分配给项目。目标是找到一个稳定匹配，确保在给定分配方案下，不存在学生和导师能够通过偏离当前安排、转而参与其他项目而共同获益。我们探讨了SPA-S的结构特性，并刻画了任意SPA-S实例的稳定匹配集合。我们证明，与经典稳定婚姻问题（SM）和医院-住院医师匹配问题（HR）类似，SPA-S中所有稳定匹配的集合构成一个分配格。在该格中，学生最优稳定匹配和导师最优稳定匹配分别代表最小元和最大元。最后，我们通过图示在SPA-S框架中引入元旋转概念，展示其如何刻画稳定匹配间的关联关系。这些新颖的结构性发现为开发高效算法解决SPA-S中若干关于稳定匹配的开放性问题奠定了基础。