Super-stability in the Student-Project Allocation Problem with Ties

TL;DR

This paper introduces a polynomial-time algorithm for finding super-stable matchings in the generalized Student-Project Allocation problem with ties, and empirically analyzes the likelihood of such matchings under various tie configurations.

Contribution

It presents the first efficient algorithm for super-stable matchings in SPA-ST and explores the impact of ties on the existence of these matchings.

Findings

Super-stable matchings are more likely when ties are only in lecturers' preferences.

The algorithm runs in linear time relative to preference list length.

Super-stable matchings are often elusive with ties in both students and lecturers.

Abstract

The Student-Project Allocation problem with lecturer preferences over Students (SPA-S) involves assigning students to projects based on student preferences over projects, lecturer preferences over students, and the maximum number of students that each project and lecturer can accommodate. This classical model assumes that each project is offered by one lecturer and that preference lists are strictly ordered. Here, we study a generalisation of SPA-S where ties are allowed in the preference lists of students and lecturers, which we refer to as the Student-Project Allocation problem with lecturer preferences over Students with Ties (SPA-ST). We investigate stable matchings under the most robust definition of stability in this context, namely super-stability. We describe the first polynomial-time algorithm to find a super-stable matching or to report that no such matching exists, given an instance of SPA-ST. Our algorithm runs in $O(L)$ time, where $L$ is the total length of all the preference lists. Finally, we present results obtained from an empirical evaluation of the linear-time algorithm based on randomly-generated SPA-ST instances. Our main finding is that, whilst super-stable matchings can be elusive when ties are present in the students' and lecturers' preference lists, the probability of such a matching existing is significantly higher if ties are restricted to the lecturers' preference lists.