Optimal Instructional Sequencing Complexity Explored.

The challenge of determining the optimal sequence for teaching concepts linked by prerequisite dependencies is a fundamental problem in education and training. This paper delves into the computational complexity of this "instructional sequencing" problem, framing it as a stochastic shortest-path problem where the success of learning a concept is probabilistic.A key finding is that the inherent stochasticity of learning can be precisely eliminated. The problem can be reduced to an equivalent deterministic shortest-path problem on a lattice of prerequisite order ideals, preserving both optimal values and actions. This reduction simplifies the problem by removing stochastic complexity, yet the combinatorial complexity persists.The research demonstrates that optimal sequencing remains NP-hard even under simplified conditions, such as the absence of prerequisite edges or uniform costs. However, the hardness is not universal; polynomial-time exact dynamic programming solutions are possible when transfer preferences remain acyclic with prerequisites or when prerequisite width is fixed. A diagnostic tool, mΔ, is introduced to bound the value of sequencing before optimization, identifying scenarios where optimization yields little benefit.