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CS-120 · computer science · mathematics · Computational statistics

Exact uniform sampling of arbitrary-margin contingency tables

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For all nonnegative integer row and column margins of equal total N, is exact uniform sampling of nonnegative integer contingency tables possible with independent unbiased random bits, almost-sure termination and polynomial expected bit time in dimensions and log(N+1)? Is total-variation error $2^{-k}$ achievable in bounded polynomial time in those parameters and k?

Short English statement

For all nonnegative integer row and column margins of equal total N, is exact uniform sampling of nonnegative integer contingency tables possible with independent unbiased random bits, almost-sure termination and polynomial expected bit time in dimensions and log(N+1)? Is total-variation error $2^{-k}$ achievable in bounded polynomial time in those parameters and k?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-120-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-120

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 115; 2026-09-24).

Exact uniform sampling of arbitrary-margin contingency tables

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 115; 2026-09-24), the asserted result is that arbitrary-margin nonnegative integer contingency tables admit exact uniform sampling with almost-sure termination and polynomial expected bit cost.

Mathematical statement of the claimed result

Manuscript 1.
For arbitrary nonnegative integer margins $r\in\mathbb Z_{\ge0}^m$, $c\in\mathbb Z_{\ge0}^n$ with equal total $N$, let \[ \Omega(r,c)=\{A\in\mathbb Z_{\ge0}^{m\times n}:\sum_jA_{ij}=r_i,\ \sum_iA_{ij}=c_j\}. \] The claim gives a bounded-time sampler whose output law is within $2^{-k}$ total variation of uniform for every supplied integer $k\ge1$, with cost polynomial in $m,n,\log(N+1),k$. It also gives an exact-uniform sampler terminating almost surely with expected bit time polynomial in $m,n,\log(N+1)$. Both use unbiased bits and are uniform across all margins, including zeros, with no balance or fixed-dimension assumption.

Outline of the argument

For manuscript 1 (family 115; 2026-09-24), small margins are explicit states while large completions are handled by a padded auxiliary chain. Signature and transport bounds give a bounded approximate sampler; tabulation of its dyadic output law enables a rare exhaustive residual draw that corrects the law exactly.

Formal verification

For manuscript 1 (family 115; 2026-09-24), OpenAI lists Lean statements for this assertion: arbitrary-margin nonnegative integer contingency tables admit exact uniform sampling with almost-sure termination and polynomial expected bit cost.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are ContingencyTables.lean.

Their named declarations include OAI.ContingencyTables.boundedSampling, OAI.ContingencyTables.counting, OAI.ContingencyTables.exactSampling.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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