OpenAI claim summary for CS-135
computer sciencemathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
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 113; 2026-09-23).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
FPRAS for perfect matchings in general graphs
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 113; 2026-09-23), the asserted result is that perfect matchings in arbitrary simple undirected graphs admit an FPRAS with polynomial work on every tape and certain zero output when no matching exists.Mathematical statement of the claimed result
Manuscript 1.
For every explicitly encoded finite general graph $G$, let $Z(G)$ count its perfect matchings. Given rational $0<\varepsilon<1$ and $0<\delta<1/2$, the claimed uniform classical randomized algorithm returns $\widehat Z\in\mathbb Q_{\ge0}$ with \[ \Pr[(1-\varepsilon)Z(G)\le\widehat Z\le(1+\varepsilon)Z(G)]\ge1-\delta. \] It outputs zero surely if $Z(G)=0$, and its worst-case bit cost is polynomial in the complete input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$; bipartiteness is not required.Outline of the argument
For manuscript 1 (family 113; 2026-09-23), weighted matching laws are enlarged to expose local partition ratios as marked edges. Subdivision controls unmatched-vertex costs, and alternating overlays plus a two-coordinate energy inequality bound the chain variance needed for annealing and counting.Formal verification
For manuscript 1 (family 113; 2026-09-23), OpenAI lists Lean statements for this assertion: perfect matchings in arbitrary simple undirected graphs admit an FPRAS with polynomial work on every tape and certain zero output when no matching exists.The selected statement includes the clause used for this problem.
The linked family Comparator checks are BinaryMatching.lean, MatchingEntropy.lean, MatchingEntropyBounds.lean, MatchingFPRAS.lean, SingletonLoopMatching.lean, TriangleFace.lean.
Their named declarations include OAI.BinaryMatching.deterministic_approximate_counting, OAI.LoopMatching.fullEndpoint, OAI.MatchingEntropy.entropy_main, OAI.MatchingEntropyBounds.Refined.pointwise_entropy, OAI.MatchingFPRAS.thm_main, OAI.TriangleFace198.triangle_expansion_face.
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
- OpenAI, A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs, 2026-09-23; citation key
OAI:A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 113: Lean scope.
- Comparator BinaryMatching.json.
- Comparator BinaryMatching.lean.
- Comparator MatchingEntropy.json.
- Comparator MatchingEntropy.lean.
- Comparator MatchingEntropyBounds.json.
- Comparator MatchingEntropyBounds.lean.
- Comparator MatchingFPRAS.json.
- Comparator MatchingFPRAS.lean.
- Comparator SingletonLoopMatching.json.
- Comparator SingletonLoopMatching.lean.
- Comparator TriangleFace.json.
- Comparator TriangleFace.lean.
- OpenAI catalogue entry.