CS-165 · computer science · mathematics · Optimization

Polynomial-time three-machine unit-job scheduling

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

For any explicitly listed finite acyclic precedence graph on unit-length nonpreemptive jobs, can one deterministic polynomial-bit-time algorithm produce a minimum-makespan schedule on three identical machines and decide every binary deadline exactly?

Short English statement

Sources

S1 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-165-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for CS-165

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 124; 2026-09-24).

Polynomial-time three-machine unit-job scheduling

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 124; 2026-09-24), the asserted result is that precedence-constrained unit jobs on three identical machines admit an exact deterministic polynomial-time scheduling algorithm.

Mathematical statement of the claimed result

Manuscript 1.

For every explicitly listed finite precedence DAG on $n\ge1$ nonpreemptive unit-duration jobs and three identical machines, the claimed uniform deterministic algorithm produces a schedule with globally minimum makespan. With a supplied integer deadline $1\le T\le n$, it decides feasibility exactly and constructs a feasible schedule in the affirmative case. If $L$ is total binary input length, both tasks obey \[ \operatorname{time}\le O((L+2)^{150020}) \] on a deterministic multitape Turing machine.

Outline of the argument

For manuscript 1 (family 124; 2026-09-24), feasible schedules are reorganized into recursively marked intervals. Global source, sink, and boundary predicates retain only bounded lists after exchanges and pruning, so a dynamic program enumerates polynomially many descriptions while preserving makespan feasibility.

Formal verification

For manuscript 1 (family 124; 2026-09-24), OpenAI lists Lean statements for this assertion: precedence-constrained unit jobs on three identical machines admit an exact deterministic polynomial-time scheduling algorithm.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are ThreeMachine.lean.

Their named declarations include OAI.ThreeMachine.main_theorem.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

OpenAI, A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling, 2026-09-24; citation key OAI:A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
Family 124: Lean scope.
Comparator ThreeMachine.json.
Comparator ThreeMachine.lean.
OpenAI catalogue entry.

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Record details

Subjects: Optimization
Collection: Top Open Problems
Computer Science rank: 165
Mathematics rank: 3007