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Structural aspects of the Student Project Allocation problem

Peace Ayegba, Sofiat Olaosebikan, David Manlove

TL;DR

This paper studies the SPA-S problem, a three-party stable matching framework with students, projects, and lecturers under capacity constraints. It develops a structural analysis of stable matchings by introducing a student-oriented dominance relation and proving that the set of stable matchings forms a finite distributive lattice, with the student-optimal and lecturer-optimal matchings serving as the lattice's bottom and top elements. The authors prove this lattice property via three new lemmas, refined meet and join constructions, and the notion of meta-rotations to compactly represent all stable matchings. These insights enable algorithmic opportunities for enumerating stable matchings, identifying stable pairs, and computing egalitarian or other optimized matchings in SPA-S, while outlining open questions for SPA-S extensions such as spa-st with ties.

Abstract

We study the Student Project Allocation problem with lecturer preferences over Students (SPA-S), which involves the assignment of students to projects based on student preferences over projects, lecturer preferences over students, and capacity constraints on both projects and lecturers. The goal is to find a stable matching that ensures no student and lecturer can mutually benefit by deviating from a given assignment to form an alternative arrangement involving some project. We explore the structural properties of SPA-S and characterise the set of stable matchings for an arbitrary SPA-S instance. We prove that, similar to the classical Stable Marriage problem (SM) and the Hospital Residents problem (HR), the set of all stable matchings in SPA-S forms a distributive lattice. In this lattice, the student-optimal and lecturer-optimal stable matchings represent the minimum and maximum elements, respectively. Finally, we introduce meta-rotations in the SPA-S setting using illustrations, demonstrating how they capture the relationships between stable matchings. These novel structural insights paves the way for efficient algorithms that address several open problems related to stable matchings in SPA-S.

Structural aspects of the Student Project Allocation problem

TL;DR

This paper studies the SPA-S problem, a three-party stable matching framework with students, projects, and lecturers under capacity constraints. It develops a structural analysis of stable matchings by introducing a student-oriented dominance relation and proving that the set of stable matchings forms a finite distributive lattice, with the student-optimal and lecturer-optimal matchings serving as the lattice's bottom and top elements. The authors prove this lattice property via three new lemmas, refined meet and join constructions, and the notion of meta-rotations to compactly represent all stable matchings. These insights enable algorithmic opportunities for enumerating stable matchings, identifying stable pairs, and computing egalitarian or other optimized matchings in SPA-S, while outlining open questions for SPA-S extensions such as spa-st with ties.

Abstract

We study the Student Project Allocation problem with lecturer preferences over Students (SPA-S), which involves the assignment of students to projects based on student preferences over projects, lecturer preferences over students, and capacity constraints on both projects and lecturers. The goal is to find a stable matching that ensures no student and lecturer can mutually benefit by deviating from a given assignment to form an alternative arrangement involving some project. We explore the structural properties of SPA-S and characterise the set of stable matchings for an arbitrary SPA-S instance. We prove that, similar to the classical Stable Marriage problem (SM) and the Hospital Residents problem (HR), the set of all stable matchings in SPA-S forms a distributive lattice. In this lattice, the student-optimal and lecturer-optimal stable matchings represent the minimum and maximum elements, respectively. Finally, we introduce meta-rotations in the SPA-S setting using illustrations, demonstrating how they capture the relationships between stable matchings. These novel structural insights paves the way for efficient algorithms that address several open problems related to stable matchings in SPA-S.

Paper Structure

This paper contains 16 sections, 17 theorems, 6 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contributions
  4. Structure of the paper
  5. Preliminary Definitions
  6. The Student-Project Allocation model
  7. Example.
  8. Preferences over Matchings
  9. Student Preferences over Matchings
  10. Lecturer Preferences over Matchings
  11. Dominance relation
  12. Structural Properties of Stable Matchings
  13. Stable Matchings in spa-s Form a Distributive Lattice
  14. Example
  15. Conclusion and Open Problems
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\mathcal{M}$ denote the set of all stable matchings in a given instance of spa-s. Then:

Figures (5)

  • Figure 1: An instance $I_1$ of spa-s
  • Figure 2: An illustration of the sequence of students generated in Lemma \ref{['lem:samelecturer1']}, with $(s_r, p_r) \in M$ and $(s_r, p_{r-1}) \in M'$ for all $r \geq 2$
  • Figure 3: A spa-s instance illustrating the infinite sequence of students generated in Lemma \ref{['lem:s-prefers-l-prefers1']}, where $(s_t, p_t) \in M'$ and $(s_t, p_{t-1}) \in M$.
  • Figure 4: An instance $I_3$ of spa-s
  • Figure 5: Lattice structure for $I_3$

Theorems & Definitions (38)

  • Theorem 2.1: Unpopular Projects Theorem AIM2007O2020
  • Definition 2.1: Student-oriented dominance relation
  • Definition 2.2: Lecturer-oriented dominance
  • Proposition 2.1
  • proof
  • Definition 2.3: Distributive lattice GI1989
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 28 more