A 3/2-approximation algorithm for the Student-Project Allocation problem

TL;DR

This paper introduces a linear-time 3/2-approximation algorithm for the NP-hard maximum stable matching problem in the Student-Project Allocation with lecturer preferences and ties, achieving near-optimal solutions efficiently.

Contribution

It presents the first efficient approximation algorithm for MAX SPA-ST with ties, outperforming the theoretical bound in practice and providing an exact Integer Programming model.

Findings

The approximation algorithm consistently finds solutions within 92% of optimal.

The algorithm surpasses the 3/2 approximation bound in all tested cases.

Experimental results show the algorithm's practical effectiveness for large instances.

Abstract

The Student-Project Allocation problem with lecturer preferences over Students (SPA-S) comprises three sets of agents, namely students, projects and lecturers, where students have preferences over projects and lecturers have preferences over students. In this scenario we seek a stable matching, that is, an assignment of students to projects such that there is no student and lecturer who have an incentive to deviate from their assignee/s. We study SPA-ST, the extension of SPA-S in which the preference lists of students and lecturers need not be strictly ordered, and may contain ties. In this scenario, stable matchings may be of different sizes, and it is known that MAX SPA-ST, the problem of finding a maximum stable matching in SPA-ST, is NP-hard. We present a linear-time 3/2-approximation algorithm for MAX SPA-ST and an Integer Programming (IP) model to solve MAX SPA-ST optimally. We compare the approximation algorithm with the IP model experimentally using randomly-generated data. We find that the performance of the approximation algorithm easily surpassed the 3/2 bound, constructing a stable matching within 92% of optimal in all cases, with the percentage being far higher for many instances.