The Plan Language of a Curriculum: A Formal Model and the Complexity of Degree Planning

2026-10-02

AI Summary

This study addresses the computational complexity challenges introduced by course prerequisites and credit constraints in degree planning. It models courses as a formal language and conducts formal analysis using monotone Boolean formulas, covering constraints, and combinatorial optimization theory. Furthermore, this work proposes the concept of “disjunctive relaxation” to refine standard complexity measures, thereby decoupling the sources of hardness associated with graduation time and total credit load. The findings reveal that most curricula exhibit a purely conjunctive structure and demonstrate that capacity limitations, rather than prerequisite logic, constitute the primary bottleneck constraining graduation time. Additionally, the relevant datasets are released as open source to facilitate future research.

Abstract

We model an academic curriculum as a generator of a language of feasible study plans: prerequisites are monotone Boolean formulas in conjunctive normal form, degree requirements are credit-threshold covering constraints, and a study plan is a sequence of terms bounded by a per-term credit capacity. Within this model, we settle the complexity of the two natural planning objectives, the number of terms to a degree and the total credit load, and we isolate the structural commitment responsible for each source of hardness. Time to degree is polynomial whenever the per-term capacity is unbounded, for arbitrary disjunctive prerequisites and arbitrary electives, so disjunction never contributes to its hardness, yet it becomes strongly NP-hard as soon as capacity binds, even without any prerequisite. Load is complementary: disjunction and overlapping electives are each strongly NP-hard in isolation and their complexity does not depend on capacity, while load is polynomial on the conjunctive, mandatory fragment. The two objectives therefore have disjoint sources of hardness. We show that the delay-factor component of the standard curricular-complexity metric is a polynomially computable upper bound on time to degree, exact on the conjunctive fragment and loose elsewhere by a quantity we name the disjunctive slack, and we prove that program subsumption is coNP-complete and consensus prerequisite recovery is NP-complete. Instantiating the model on a corpus of twenty-two universities, we find that 88 percent of prerequisite-bearing courses are purely conjunctive and that capacity, not prerequisite logic, is the operative constraint on time to degree. The curriculum corpus is openly available (https://doi.org/10.5281/zenodo.22334674), and the analysis and figure-generation code accompany the paper.

Problem

• curriculum planning
• computational complexity
• degree requirements
• study plan optimization
• prerequisite constraints

Innovation

• curriculum modeling
• computational complexity
• degree planning
• Boolean formulas
• NP-hard