Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Course Allocation with Credits via Stable Matching

José Rodríguez, David Manlove

TL;DR

The paper introduces a formal model for course allocation where courses carry different credit values and may have conflicts, and studies four stability notions—pair stability, pair-size stability, coalition stability, and first-coalition stability. It develops a polynomial-time algorithm to find pair-size-stable matchings in the arbitrary-preference setting and shows existence guarantees under master lists via Serial Dictatorship, with uniqueness. It also thoroughly maps the computational complexity of verifying stability and finding stable or maximum-size stable matchings, proving NP-hardness for most definitions and identifying polynomial cases primarily when master lists are present. The analysis extends to lower quotas and their impact, showing that while some problems remain tractable under master lists, many stability-related optimization problems become NP-hard with quotas or closures, highlighting fundamental limits and guiding future research on stability in heterogeneous-credit allocation systems.

Abstract

In the {\sc Course Allocation} problem, there are a set of students and a set of courses at a given university. University courses may have different numbers of credits, typically related to different numbers of learning hours, and there may be other constraints such as courses running concurrently. Our goal is to allocate the students to the courses such that the resulting matching is stable, which means that no student and course(s) have an incentive to break away from the matching and become assigned to one another. We study several definitions of stability and for each we give a mixture of polynomial-time algorithms and hardness results for problems involving verifying the stability of a matching, finding a stable matching or determining that none exists, and finding a maximum size stable matching. We also study variants of the problem with master lists of students, and lower quotas on the number of students allocated to a course, establishing additional complexity results in these settings.

Course Allocation with Credits via Stable Matching

TL;DR

The paper introduces a formal model for course allocation where courses carry different credit values and may have conflicts, and studies four stability notions—pair stability, pair-size stability, coalition stability, and first-coalition stability. It develops a polynomial-time algorithm to find pair-size-stable matchings in the arbitrary-preference setting and shows existence guarantees under master lists via Serial Dictatorship, with uniqueness. It also thoroughly maps the computational complexity of verifying stability and finding stable or maximum-size stable matchings, proving NP-hardness for most definitions and identifying polynomial cases primarily when master lists are present. The analysis extends to lower quotas and their impact, showing that while some problems remain tractable under master lists, many stability-related optimization problems become NP-hard with quotas or closures, highlighting fundamental limits and guiding future research on stability in heterogeneous-credit allocation systems.

Abstract

In the {\sc Course Allocation} problem, there are a set of students and a set of courses at a given university. University courses may have different numbers of credits, typically related to different numbers of learning hours, and there may be other constraints such as courses running concurrently. Our goal is to allocate the students to the courses such that the resulting matching is stable, which means that no student and course(s) have an incentive to break away from the matching and become assigned to one another. We study several definitions of stability and for each we give a mixture of polynomial-time algorithms and hardness results for problems involving verifying the stability of a matching, finding a stable matching or determining that none exists, and finding a maximum size stable matching. We also study variants of the problem with master lists of students, and lower quotas on the number of students allocated to a course, establishing additional complexity results in these settings.

Paper Structure

This paper contains 14 sections, 24 theorems, 23 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Related work
  4. Our contributions
  5. Structure of the paper
  6. Model
  7. Verifying stability
  8. Finding a stable matching
  9. Arbitrary course preferences
  10. Master lists
  11. Finding a maximum size stable matching
  12. No lower quotas
  13. With lower quotas
  14. Conclusion

Key Result

Proposition 1

Pair stability implies first-coalition stability, which implies coalition stability, which implies pair-size stability.

Figures (8)

  • Figure 1: Example of a matching in a ca instance, highlighted in green, and a blocking pair, highlighted in red.
  • Figure 2: Example of a pair-size-stable matching in a ca instance, highlighted in green. A subset of courses that $s_1$ may prefer over $c_3$ is shown in red.
  • Figure 3: Example of a coalition-stable matching in a ca instance, highlighted in green. A subset of courses that $s_1$ may prefer over $c_3$ is shown in red.
  • Figure 4: Instance of ca. This instance is derived from an hrs instance due to McDermid and Manlove mcdermid_keeping_2010.
  • Figure 5: Preference list of $s_1$, with $T(s_1) = 2$. $O(c_1) = O(c_3) = 1$, and $O(c_2) = 2$.
  • ...and 3 more figures

Theorems & Definitions (55)

  • Definition 1
  • Definition 2: utture_student_2019
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • ...and 45 more