发表机构
University of California, Davis; Minjiang University; Liaoning Normal University（加州大学戴维斯分校; 闽江学院; 辽宁师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将教学排序归约为随机最短路径问题，证明其随机性可消除且最优排序仍为NP难，提出诊断量$m\boldsymbol{\triangle}$，并在课程交互数据上验证了易处理与挑战性 regime，A*算法可高效求解挑战性实例。
AI 中文摘要
当学生必须学习通过先决条件依赖关系关联的概念时，教学顺序何时重要，找到最优顺序的成本是多少？我们将教学排序研究为一个随机最短路径问题，其中尝试掌握某个概念的成功概率依赖于当前状态，且失败不会改变学习者的状态。我们首先证明该随机性可被精确消除：该问题可归约为先决条件序理想格上的确定性最短路径问题，且保留最优值与最优动作。该归约消除了随机复杂度，但未消除组合复杂度：最优排序仍是NP难问题——通过从锦标赛反馈弧集归约得到——即使在无先决条件边、单位成本、均匀二元非负迁移且成功概率至少为1/2的情况下，该问题仍具NP难性。该难性并非均匀存在：当可实现的迁移偏好与先决条件共同构成无环图时，该残差联合图的任意拓扑序均为最优，且固定先决条件宽度可得到多项式时间的精确动态规划算法。一个可计算的诊断量$m\boldsymbol{\triangle}$可界定优化前排序的价值。在一门入门级CS课程的70893次交互数据上，该诊断量证实了双重易处理 regime——优化价值小且搜索空间小；而构造的迁移实例则实现了具挑战性的 regime，其中近视排序会产生大的遗憾，但采用一致启发式的精确A*算法在该实例族上仅需扩展线性数量的状态。
英文摘要
When a student must learn concepts connected by prerequisite dependencies, when does the order of instruction matter, and what does it cost to find the best one? We study instructional sequencing as a stochastic shortest-path problem in which attempting a concept succeeds with a state-dependent probability and failure leaves the learner state unchanged. We first prove that this stochasticity can be eliminated exactly: the problem collapses to a deterministic shortest-path problem on the lattice of prerequisite order ideals, preserving optimal values and actions. The collapse removes stochastic complexity but not combinatorial complexity: optimal sequencing remains NP-hard -- via reduction from feedback arc set in tournaments -- even with no prerequisite edges, unit costs, uniform binary nonnegative transfer, and success probabilities at least $1/2$. Hardness is not uniform: when realizable transfer preferences remain jointly acyclic with the prerequisites, any topological order of the residual joint graph is optimal, and fixed prerequisite width yields polynomial-time exact dynamic programming. A computable diagnostic, $mΔ$, bounds the value of sequencing before optimization. On 70,893 interactions from an introductory CS course, the diagnostic certifies a doubly easy regime -- little value to optimize and little space to search -- while constructed transfer instances realize the challenging regime, where myopic sequencing suffers large regret yet exact A* with a consistent heuristic expands only linearly many states on that family.
Comments11 pages, 1 figure, 1 table. Equal contribution among Y. Chen, J. Jiang, and T. Shi (alphabetical order)
