๐ค AI Summary
This study addresses the problem of finding the optimal teaching sequence that minimizes learning cost in scenarios with prerequisite dependencies. The problem is modeled as a stochastic shortest path problem, and we propose an exact reduction based on lattice theory that transforms it into a deterministic shortest path problem, revealing that the computational difficulty stems not from stochasticity but from the combinatorial complexity inherent in the dependency structure. We theoretically prove the problem to be NP-hard; however, by leveraging dynamic programming, A* search, and feedback arc set reductions, we identify a โdoubly simpleโ regime in real-world course data where A* efficiently solves instances with state-space size linear in the number of concepts. A computable diagnostic metric, \( m\Delta \), further enables practical assessment of instance hardness.
๐ Abstract
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, $mฮ$, 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.