OpenAI · 124-01 · 2026-09-24 · Theoretical computer science
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling
preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026
Family 124. Polynomial-time scheduling on three identical machines.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 124: Polynomial-time scheduling on three identical machines. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.