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.

翻译：本研究探讨了带有导师对学生偏好的学生项目分配问题（SPA-S），这是经典稳定婚姻与医院住院医师匹配问题的扩展模型。在该模型中，学生对于项目具有偏好，每个项目由单一导师提供，而导师则对学生具有偏好。目标是计算一个稳定匹配，即将学生分配至项目（从而分配至导师）的方案，使得任何学生或导师均无动机偏离当前分配。尽管该问题源于高校情境，但同样出现在许多由具有自身偏好的主体提供有限资源的分配场景中，例如无线网络。通过发展元旋转理论——对稳定婚姻问题中经典旋转概念的推广，我们建立了SPA-S中稳定匹配集合的新结构性质。每个元旋转对应于稳定匹配格中，将一个稳定匹配转化为另一个稳定匹配的最小变更集合。所有元旋转按其优先关系排序构成元旋转偏序集。我们证明了稳定匹配集合与元旋转偏序集的闭子集之间存在一一对应关系。通过构建该结构理论，我们为设计高效算法奠定了基础，这些算法可用于枚举与计数稳定匹配，以及计算其他最优稳定匹配（例如平等主义匹配或最小成本匹配），这些在SPA-S中尚未被系统研究。