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CS-122 · computer science · mathematics · Algorithms and data structures · Computational statistics

FPRAS for cell-bounded contingency tables

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For arbitrary binary nonnegative row and column margins and binary upper bounds on every cell, is there one randomized relative counting scheme using polynomial worst-case bit time in total input length, $\varepsilon ^{-1}$ and $\log (\delta ^{-1})$, with failure probability $\delta $ and certain zero output for emptiness, without a counting or sampling oracle?

Short English statement

For arbitrary binary nonnegative row and column margins and binary upper bounds on every cell, is there one randomized relative counting scheme using polynomial worst-case bit time in total input length, $\varepsilon ^{-1}$ and $\log (\delta ^{-1})$, with failure probability $\delta $ and certain zero output for emptiness, without a counting or sampling oracle?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-122-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-122

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).

FPRAS for cell-bounded 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 binary-encoded contingency tables with individual cell bounds admit approximate counting with relative-error confidence guarantees and polynomial work on every random execution.

Mathematical statement of the claimed result

Manuscript 1.
Given nonnegative integer row margins $r_i$, column margins $c_j$ with equal total, and cell capacities $b_{ij}$, all in binary, define \[ \Omega=\{A\in\mathbb Z_{\ge0}^{m\times n}:\ \sum_jA_{ij}=r_i,\ \sum_iA_{ij}=c_j,\ A_{ij}\le b_{ij}\},\quad Z=|\Omega|. \] For rational $0<\varepsilon,\delta<1$, the claimed randomized algorithm uses only unbiased bits and returns $\widehat Z\in\mathbb Q_{\ge0}$ with \[ \Pr[(1-\varepsilon)Z\le\widehat Z\le(1+\varepsilon)Z]\ge1-\delta. \] If $Z=0$ the output is always zero. Every execution has bit cost bounded by one polynomial in total binary input length $L$, $\varepsilon^{-1}$ and $\log\delta^{-1}$; no sampling or counting oracle is assumed.

Outline of the argument

For manuscript 1 (family 115; 2026-09-24), softened integral signatures and balanced summation extend weights to defect states. A transport inequality controls the annealing chain, and bin-based rational weight estimation with bounded fair-bit simulation provides a finite implementation.

Formal verification

For manuscript 1 (family 115; 2026-09-24), OpenAI lists Lean statements for this assertion: binary-encoded contingency tables with individual cell bounds admit approximate counting with relative-error confidence guarantees and polynomial work on every random execution.

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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