Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. (Lean)
Manuscript
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling
September 24, 2026 · source · cite as \cite{OAI:A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026}
We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions. Global boundary conditions and simplification of inherited information keep the descriptions bounded throughout the decomposition.
Result 124 · Theoretical computer science. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md