OpenAI · 113-01 · 2026-09-23 · Theoretical computer science
A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs
preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026
Family 113. Approximate counting and the perfect-matching entropy conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.
We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, , and .
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 113: Approximate counting and entropy of perfect matchings. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.