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A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs

Theorem 1.1: a uniform classical randomized algorithm that, for any finite simple undirected graph $G$ and rationals $0<\varepsilon<1$ , $0<\delta<1/2$ , returns $\widehat Z\ge0$ with $\Pr[(1-\varepsilon)Z(G)\le\widehat Z\le(1+\varepsilon)Z(G)]\ge1-\delta$ , returns 0 with certainty when $Z(G)=0$ , and runs in worst-case bit time polynomial in the input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$ . Corollaries give counting schemes for edge sets with prescribed degrees and for matchings of a given size. It is for unweighted simple graphs: it does not claim an FPRAS for hafnians with arbitrary nonnegative weights, and the polynomial's degree is not optimised.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs · FPRAS for perfect matchings in general graphs

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the question as Jerrum and Sinclair raised it (Section 7(i) of their 1989 paper, as the manuscript cites it). The proof was not refereed. Lean: the release's Comparator challenge MatchingFPRAS (theorem OAI.MatchingFPRAS.thm_main, solution module OAI.Combinatorics.MatchingCount.Main, both present at the pinned commit) is not in the release's formalization catalogue (lean/formalization.yaml); it is reached through lean/docs/113.md. Its statement was read here: a single finite randomized Post-Turing machine, on an explicit binary encoding of any finite simple graph and rationals $0<\varepsilon<1$ , $0<\delta<1/2$ , halts on every random tape within $C(\text{input length}+\lceil\varepsilon^{-1}\rceil+\lceil\log_2\delta^{-1}\rceil+1)^d$ steps with a nonnegative rational, outputs 0 on every tape when there is no perfect matching, and is within relative error $\varepsilon$ on at least a $1-\delta$ fraction of tapes. That is the headline claim. The development was not rebuilt here and the statement has not been independently audited.

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Same family: A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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The Anari-Oveis Gharan-Vinzant perfect-matching entropy conjecture

witness · pending

paper link supplied by the VibeMathed registry.

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Lean Comparator challenge MatchingFPRAS (statement)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean solution module OAI.Combinatorics.MatchingCount.Main

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Jerrum and Sinclair 1989, Approximating the permanent

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge MatchingFPRAS (statement)

lean artifact · pending

Artifact ↗

Lean solution module OAI.Combinatorics.MatchingCount.Main

lean artifact · pending

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Lineage and corrections

A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs parent of this event

This event attributed to Unreleased internal OpenAI model

Lean solution module OAI.Combinatorics.MatchingCount.Main evidence for this event

Same family: A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs evidence for this event

Jerrum and Sinclair 1989, Approximating the permanent evidence for this event

Lean Comparator challenge MatchingFPRAS (statement) evidence for this event

VibeMathed record: A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs evidence for this event

The Anari-Oveis Gharan-Vinzant perfect-matching entropy conjecture evidence for this event

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