# Profile-Based Optimal Matchings in the Student/Project Allocation Problem

## Abstract

In the *Student/Project Allocation problem* (spa) we seek to assign students to individual or group projects offered by lecturers. Students provide a list of projects they find acceptable in order of preference. Each student can be assigned to at most one project and there are constraints on the maximum number of students that can be assigned to each project and lecturer. We seek matchings of students to projects that are optimal with respect to *profile*, which is a vector whose *r*th component indicates how many students have their *r*th-choice project. We present an efficient algorithm for finding a*greedy maximum matching* in the spa context – this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a *generous maximum matching* – this is a matching whose reverse profile is lexicographically minimum. Our algorithms involve finding optimal flows in networks. We demonstrate how this approach can allow for additional constraints, such as lecturer lower quotas, to be handled flexibly.

## Keywords

Allocation Problem Edge Weight Generous Maximum Maximum Match Preference List## References

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