[论文解读] An Algorithmic Approach to Address Course Enrollment Challenges

Arpita Biswas, Yiduo Ke+2 more|arXiv (Cornell University)|Jan 1, 2023

Optimization and Search Problems被引用 1

一句话总结

本文提出了一种在存在时间冲突和学分上限情况下的公平且高效的课程选课算法框架，将课程调度建模为基于区间的资源分配问题。该框架提出近线性时间的近似算法，总效用的近似因子为1/2，最大-最小效用的近似因子为1/4，并通过实验验证表明，与整数规划相比，运行时间显著减少，同时性能接近最优。

ABSTRACT

Massive surges of enrollments in courses have led to a crisis in several computer science departments - not only is the demand for certain courses extremely high from majors, but the demand from non-majors is also very high. Much of the time, this leads to significant frustration on the part of the students, and getting seats in desired courses is a rather ad-hoc process. One approach is to first collect information from students about which courses they want to take and to develop optimization models for assigning students to available seats in a fair manner. What makes this problem complex is that the courses themselves have time conflicts, and the students have credit caps (an upper bound on the number of courses they would like to enroll in). We model this problem as follows. We have n agents (students), and there are "resources" (these correspond to courses). Each agent is only interested in a subset of the resources (courses of interest), and each resource can only be assigned to a bounded number of agents (available seats). In addition, each resource corresponds to an interval of time, and the objective is to assign non-overlapping resources to agents so as to produce "fair and high utility" schedules. In this model, we provide a number of results under various settings and objective functions. Specifically, in this paper, we consider the following objective functions: total utility, max-min (Santa Claus objective), and envy-freeness. The total utility objective function maximizes the sum of the utilities of all courses assigned to students. The max-min objective maximizes the minimum utility obtained by any student. Finally, envy-freeness ensures that no student envies another student’s allocation. Under these settings and objective functions, we show a number of theoretical results. Specifically, we show that the course allocation under the time conflicts problem is NP-complete but becomes polynomial-time solvable when given only a constant number of students or all credits, course lengths, and utilities are uniform. Furthermore, we give a near-linear time algorithm for obtaining a constant 1/2-factor approximation for the general maximizing total utility problem when utility functions are binary. In addition, we show that there exists a near-linear time algorithm that obtains a 1/2-factor approximation on total utility and a 1/4-factor approximation on max-min utility when given uniform credit caps and uniform utilities. For the setting of binary valuations, we show three polynomial time algorithms for 1/2-factor approximation of total utility, envy-freeness up to one item, and a constant factor approximation of the max-min utility value when course lengths are within a constant factor of each other. Finally, we conclude with experimental results that demonstrate that our algorithms yield high-quality results in real-world settings.

研究动机与目标

• 为应对计算机科学课程选课激增带来的危机，该危机源于本专业和非本专业的学生需求同时上升。
• 设计一种公平且高效的分配机制，尊重课程之间的时间冲突和学生的学分上限。
• 针对三种公平性目标进行优化：总效用、最大-最小（Santa Claus）公平性，以及无 envy（无嫉妒）性。
• 开发可扩展的算法，在运行时间上优于传统整数规划，同时保持高质量的解。
• 使用真实世界和合成数据集进行验证，证明该方法在实际应用中的可行性。

提出的方法

• 将课程选课建模为基于区间的资源分配问题，其中重叠的区间产生不可行约束。
• 通过区间图表示冲突，确保不会为任何学生分配时间重叠的课程。
• 应用近似算法，在二元评分和统一约束下，最大化总效用、最大-最小效用，并实现无嫉妒性。
• 在近线性时间的算法框架中，使用交换路径操作和网络流技术。
• 使用 NetworkX 和 Gurobi 对两种算法（算法3和算法4）进行实现与评估，以作比较。
• 采用二元评分（感兴趣或不感兴趣），并假设时间以离散时间步长进行调度。

实验结果

研究问题

• RQ1 在存在时间冲突和学分上限的情况下，能否实现总效用的常数因子近似？
• RQ2 在统一学分上限和统一效用下，最大-最小公平性目标（Santa Claus）能达到的近似比是多少？
• RQ3 能否设计一种多项式时间算法，确保在基于区间的冲突下，实现无嫉妒性（至多一个物品）？
• RQ4 所提出的算法在运行时间和效用方面与最优整数规划解相比表现如何？
• RQ5 这些算法在比测试数据集更大的真实世界实例上是否能有效扩展？

主要发现

• 在存在时间冲突的课程分配问题属于 NP-完全问题，但当学生数量或不同学分上限、课程长度和效用的数量为常数时，问题变为多项式时间可解。
• 当效用为二元时，近线性时间算法可实现总效用的1/2因子近似。
• 在统一学分上限和统一效用下，近线性时间算法可实现总效用的1/2因子近似和最大-最小效用的1/4因子近似。
• 在二元评分和课程长度处于常数因子范围内时，三种多项式时间算法可分别实现总效用的1/2因子近似、无嫉妒性（至多一个物品）以及最大-最小效用的常数因子近似。
• 实验结果表明，算法3在显著减少运行时间的同时，实现了接近最优的效用，尤其在大规模实例上表现优于基于 Gurobi 的整数规划。
• 尽管算法4在小规模实例上运行较慢，但在大规模数据集上展现出强大的可扩展性，表明其在真实世界课程选课系统中的实际可行性。