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Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins?

Theorem 1.2: for every fixed integer $c\ge0$ it is NP-hard to distinguish pairs $(I,B)$ where $I$ packs into $B$ bins from pairs where $I$ needs more than $B+c$ bins, with rational sizes all greater than $1/6$ . Corollary 1.3: a deterministic polynomial-time algorithm using at most $OPT(I)+C$ bins for an absolute constant $C$ exists if and only if P = NP. The reduction goes from vertex cover with Håstad's gap through a competing two-tree interval construction encoded in five-item bins. Not shown: a lower bound of order $\log OPT$ matching Hoberg-Rothvoss, or hardness for a fixed number of item sizes (polynomial by Goemans-Rothvoss).

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins? · Bin packing within OPT + constant

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins?

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.2 and Corollary 1.3 were read against the question as Williamson and Shmoys pose it. Theorem 1.2 makes distinguishing $B$ bins from more than $B+c$ bins NP-hard for each fixed $c$ , with sizes above $1/6$ ; Corollary 1.3 says an $OPT+C$ algorithm exists if and only if P = NP. The answer is negative unless P = NP, the standard form of a complete answer to such a question; solveType disproved is recorded on that understanding. Lean: the same formalised declaration OAI.BinPackingGap.main_results (listed in formalization.yaml) was read here; its second and third conjuncts state PackingGapNPHard c for every c (polynomial-time reductions from every NP language with completeness and soundness) and that an absolute additive algorithm exists iff P = NP, with P, NP and polynomial time defined on Mathlib's TM2 machines. That states the headline. Not rebuilt here; the hand-built complexity definitions were read but not audited.

Source ↗

OpenAI, Additive hardness and unbounded configuration gaps in bin packing (source theorem)

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof: OAI/Computability/BinPacking/Main.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean statement: ComparatorChallenges/BinPackingGap.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Williamson and Shmoys, The Design of Approximation Algorithms (2011), Chapter 17

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean proof: OAI/Computability/BinPacking/Main.lean

lean artifact · pending

Artifact ↗

Lean statement: ComparatorChallenges/BinPackingGap.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Lean statement: ComparatorChallenges/BinPackingGap.lean evidence for this event

This event attributed to Unreleased internal OpenAI model

Williamson and Shmoys, The Design of Approximation Algorithms (2011), Chapter 17 evidence for this event

Lean proof: OAI/Computability/BinPacking/Main.lean evidence for this event

VibeMathed record: Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins? evidence for this event

Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins? parent of this event

OpenAI, Additive hardness and unbounded configuration gaps in bin packing (source theorem) evidence for this event

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