An Algorithm for Strong Stability in the Student-Project Allocation Problem with Ties

We study a variant of the Student-Project Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (SPA-ST). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer $l$ such that if they decide to form a private arrangement outside of the matching via one of $l$'s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomial-time algorithm to find a strongly stable matching or to report that no such matching exists, given an instance of SPA-ST. Our algorithm runs in $O(m^2)$ time, where $m$ is the total length of the students' preference lists.