Result 118, Theoretical computer science

Bin packing and unbounded configuration-LP gaps

Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A fractional model of bin packing can underestimate the number of real bins needed by an arbitrarily large amount. The manuscript also reports a computational barrier to guaranteeing a solution only a fixed number of bins above optimal.

What changes?

The manuscript reports both results for rational item sizes greater than one-sixth of a bin's capacity, limiting each bin to five items. Its configuration linear program assigns fractional weights to feasible bin contents; the integral optimum counts actual bins. The latter can exceed the former by arbitrarily large amounts. For every fixed nonnegative integer c, distinguishing instances fitting in B bins from those needing more than B plus c bins is NP-hard.

What does that help mathematicians do?

The claimed gap disproves the Modified Integer Round-Up Conjecture of Scheithauer and Terno. It rules out a universal constant correction that would turn the fractional model's value into a reliable upper bound on the true optimum. Researchers therefore cannot justify such a guarantee merely by noting that each bin holds few items. This concerns extra bins, not necessarily a large percentage error.

Are there practical applications?

The immediate value is foundational for packing algorithms and optimization guarantees. The configuration model still supplies a lower bound, but these results limit what that bound can certify. The hardness claim also rules out polynomial-time algorithms guaranteeing any fixed additive error unless P equals NP; it does not rule out useful heuristics or guarantees for narrower families of instances.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Additive hardness and unbounded configuration gaps in bin packing

September 24, 2026 26 pages Main result formalized in Lean

We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than B+cB+c bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026,
  author = {{OpenAI}},
  title = {{Additive hardness and unbounded configuration gaps in bin packing}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026.pdf}{OAI:Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/118.md.

Bin packing and unbounded configuration-LP gaps

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized results rule out a universal additive bound for the configuration linear program in bin packing. For every integer c≥0c\ge0, there are an integer BB and a rational instance with 5B5B items whose individual-copy and size-type configuration-LP values both equal BB, but whose integral optimum is greater than B+cB+c. Distinguishing a packing in BB bins from the absence of one in B+cB+c bins is NP-hard for each fixed cc. A deterministic polynomial-time algorithm with a fixed absolute additive allowance exists exactly when P=NPP=NP; under that equality, an optimal algorithm is constructed.

Comparator links

Result Comparator statement
Unbounded configuration gaps and additive hardness BinPackingGap.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.