An Integer Programming Approach to the Student-Project Allocation Problem with Preferences over Projects

TL;DR

This paper introduces an integer programming model to find optimal maximum stable matchings in the Student-Project Allocation problem with preferences, and compares its solutions to existing approximation algorithms.

Contribution

It presents a novel integer programming approach for solving MAX-SPA-P optimally and empirically evaluates its performance against known approximation algorithms.

Findings

The 3/2-approximation algorithm produces stable matchings close to the maximum size.

The IP model effectively finds optimal solutions for MAX-SPA-P.

Approximation algorithms perform well in practice compared to the optimal solutions.

Abstract

The Student-Project Allocation problem with preferences over Projects (SPA-P) involves sets of students, projects and lecturers, where the students and lecturers each have preferences over the projects. In this context, we typically seek a stable matching of students to projects (and lecturers). However, these stable matchings can have different sizes, and the problem of finding a maximum stable matching (MAX-SPA-P) is NP-hard. There are two known approximation algorithms for MAX-SPA-P, with performance guarantees of 2 and 3/2. In this paper, we describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally. Following this, we present results arising from an empirical analysis that investigates how the solution produced by the approximation algorithms compares to the optimal solution obtained from the IP model, with respect to the size of the stable matchings constructed, on instances that are both randomly-generated and derived from real datasets. Our main finding is that the 3/2-approximation algorithm finds stable matchings that are very close to having maximum cardinality.