OpenAI reports an FPRAS for common integer polymatroid bases

With exact rank-value oracles, the number of common integer bases of two equal-rank polymatroids can be approximated by an FPRAS whose oracle calls and non-oracle bit work are polynomial on every run in ground-set size, binary input length, inverse relative error, and log inverse failure probability.

Sources for this record

What was already known

Common matroid-base counting and explicit expansion of polymatroid capacities do not directly give a binary-input algorithm: labelled element copies can be exponentially numerous and overcount occupancy vectors.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Notable

Extends randomized approximate counting to common integer polymatroid bases with binary capacities.

Limitations

Each integer vector is counted once; the total rank and capacities are binary encoded. The linked Lean scope checks common bases of two ordinary matroids, not this polymatroid theorem; a completed check of this claim remains undocumented.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and Lean-scope mismatch review for non-notifying local inclusion.

Read this published revision