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Three-machine unit-job scheduling in time O((L+2)^276)

Primarily AI-generated textHuman understanding: no partscs.DS — Data Structures and Algorithms

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

OpenAI recently gave a deterministic polynomial-time algorithm for P3|prec,p_j=1|C_max (problem OPEN8 of Garey and Johnson), with running time O((L+2)^150020) on a multitape Turing machine, L being the input length; the same exponent appears in the accompanying Lean statement. The exponent comes from enumerating all Boolean formulas with at most 10000 leaves, each leaf choosing its own triple of jobs. We show that every description actually used by the structural proof involves at most 21 distinct job triples and at most 3 distinct rank thresholds. Restricting the enumerated family accordingly, and using the length bound on accepting paths already proved there to limit the number of reachability passes, the same algorithm runs in time O((L+2)^276).

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. M. R. Garey and D. S. Johnson, Computers and Intractability , Freeman, 1979.
  2. OpenAI, A polynomial-time algorithm for three-machine unit-job scheduling , OpenAI Math Release preprint, September 24, 2026; Lean statement ComparatorChallenges/ThreeMachine.lean in the same release.

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026