Result 111, Theoretical computer science

One-sample matroid prophet inequalities against an almighty adversary

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted.

Lean formalization New or sharp bound

The bigger picture

Why it matters

How much can an online selector earn when it must accept or reject items before seeing what comes next? This manuscript claims that one sample per item suffices to compete with the best allowable selection made with all values known.

What changes?

The manuscript reports a rule for every finite matroid known beforehand: a system of allowable subsets with an exchange property. Values are independent and nonnegative, with finite expected offline optimum. Using one independent sample from each element's distribution, but no distribution descriptions, the rule earns in expectation at least 2 to the power minus 310 times that optimum. The guarantee holds even when an arrival-order adversary sees all samples, all actual values and the rule's entire random seed.

What does that help mathematicians do?

The claimed guarantee shows that sparse statistical information and adversarial ordering can be handled simultaneously under these constraints. It rules out the need for multiple samples per element merely to achieve some universal constant guarantee, even when the adversary knows the rule's randomness. That constant is independent of the matroid's size and the value distributions, so the claim is not limited to a particular constraint system or statistical model.

Are there practical applications?

The immediate value is foundational for online selection under combinatorial constraints: it identifies how little distributional information can suffice, even against an exceptionally informed opponent. It is not an efficiency result. No polynomial-time implementation is asserted, and the extremely small guaranteed fraction does not by itself establish a practically useful selection method.

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

One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary

September 23, 2026 27 pages

We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm's entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio 2−3102^{-310}.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026,
  author = {{OpenAI}},
  title = {{One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf}{OAI:One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/111.md.

One-sample matroid prophet inequalities against an almighty adversary

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

Scope

The formalization proves a one-sample matroid prophet inequality with expected reward at least 2−3102^{-310} times the expected offline optimum. Each element has one independent sample paired with an identically distributed nonnegative online value, and all coordinates are independent. The rule and finite seed law depend only on the labeled matroid; choices are irrevocable and feasible after every prefix. The arrival order may depend measurably on all samples, values, and the seed, and only the offline optimum must be integrable.

A supporting hidden-vector selection result for fixed nonnegative matroid weights gives expected worst-order reward at least 2−2932^{-293} times the optimum, again with a finite seed law and prefix feasibility.

Comparator links

Result Comparator statement
One-sample matroid prophet inequality against an almighty adversary MatroidProphet.lean
Hidden-vector matroid selection with worst-order guarantee MatroidSecretary.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.