{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng/","md":"https://postcutoff.com/e/2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng","date":"2026-10-19","date_precision":"day","short_title":"Counting perfect matchings in general graphs: OpenAI FPRAS, simplified proof","deck":null,"takeaway":"Approximately counting perfect matchings is easy for bipartite graphs (Jerrum–Sinclair–Vigoda, 2004) but had no polynomial-time scheme for general graphs. OpenAI's 23 Sept release (family 113) claimed a fully polynomial randomized approximation scheme for all graphs, built on a restricted Poincaré inequality for pairs of matchings, and a Lean development. On 19 Oct Chenghua Liu and Boning Meng posted 'a simplified, self-contained proof of OpenAI's FPRAS'. It keeps the same parameter bounds and avoids the original auxiliary vertices and tree constructions. The same release also claimed the Anari–Oveis Gharan–Vinzant perfect-matching entropy conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2610.26803: A simplified proof of the FPRAS for counting perfect matchings (Liu, Meng)","url":"https://arxiv.org/abs/2610.26803","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"openai/math: A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"openai/math: Entropy and Face Dimension of the Perfect Matching Polytope","url":"https://github.com/openai/math/blob/main/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"openai/math Lean scope for family 113","url":"https://github.com/openai/math/blob/main/lean/docs/113.md","type":"code","group":"primary","domain":"github.com"}],"official":4,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI"],"title":"OpenAI's FPRAS for counting perfect matchings in general graphs gets an independent simplified proof by Chenghua Liu and Boning Meng; Lean statement accepted in the community comparator run","summary":"OpenAI family 113 (\"Approximate counting and entropy of perfect matchings\", 23 Sept 2026) gives an FPRAS for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts, and proves the Anari–Oveis Gharan–Vinzant entropy conjecture for the perfect-matching polytope; the counting theorem has a Lean development (MatchingFPRAS). arXiv 2610.26803 (19 Oct) by Chenghua Liu and Boning Meng re-proves the key restricted Poincaré inequality more simply on the original vertex set and obtains the same guarantee, an independent human reading that supports the claim.","key_facts":["OpenAI family summary (openai/math): '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'","Liu–Meng abstract (verbatim): 'We give a simplified, self-contained proof of OpenAI's FPRAS for counting perfect matchings in general graphs. As in the original proof, we use a restricted Poincaré inequality for two matchings to sample weighted perfect matchings and estimate their total weight. We simplify the proof of this inequality by isolating parts of alternating cycles directly on the original vertex set … This avoids the auxiliary vertices and tree constructions used in the original proof … and yields the same FPRAS guarantee.'","How OpenAI's proof works (Liu–Meng's summary): thresholded connected components form a hierarchy; OpenAI 'use these connection thresholds to assign auxiliary weights to vertex pairs', giving completed weights within a polynomial factor of hierarchical reference weights, and compare two independent matchings through their alternating cycles","Background: the permanent/bipartite case is Jerrum–Sinclair–Vigoda (2004); whether perfect matchings in general (non-bipartite) graphs admit an FPRAS was a long-standing open question in approximate counting","Formal side: Lean scope lean/docs/113.md; the family's Lean challenge was accepted in oai-math-revised's community comparator run (26 Oct), per our family audit","Verification status: OpenAI preprint (not refereed) with Lean development, plus an independent simplified proof on arXiv (not refereed); we found no expert commentary beyond the oai-math-revised notes"],"key_numbers":[],"tags":["computer-science","algorithms","counting","perfect-matchings","markov-chains","openai-math-release","lean","independent-proof"],"science":{"field":"computer-science","subfield":"approximate counting / Markov chain Monte Carlo","problem":"FPRAS for counting perfect matchings in general graphs (bipartite case: Jerrum–Sinclair–Vigoda 2004); Anari–Oveis Gharan–Vinzant entropy conjecture","result":"Claimed FPRAS for all finite simple graphs (OpenAI), re-proved in simplified form by Liu and Meng","open_since":"","ai_system":["OpenAI internal model (unreleased, unnamed)"],"human_role":"Proof from OpenAI's model run; independent human re-proof and simplification by Liu and Meng","verification":{"key":"lean","label":"Lean-verified","text":"Lean development for the counting theorem; independent simplified proof (arXiv 2610.26803); not refereed"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nCounting perfect matchings exactly is #P-hard. For bipartite graphs, Jerrum, Sinclair and Vigoda found a randomized approximation\nscheme in 2004. For general graphs no such scheme was known. OpenAI's September release claimed one, with a proof built on a Poincaré\ninequality for pairs of matchings and a Lean formalisation.\n\nOn 19 Oct Chenghua Liu and Boning Meng posted a shorter, self-contained proof of the same algorithm's guarantee. They follow OpenAI's\nstrategy but handle the alternating cycles directly, without the auxiliary construction of the original.\n\n## Why it matters\n\nA famous algorithmic question now has a machine-checked claim and an independent human proof. Few results in OpenAI's release have this\nmuch support. The OpenAI preprint and the simplified proof are both still unrefereed.","disputed":[],"related":[{"id":"2026-10-30-ai-assisted-results-october-30-listing","url":"https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/","date":"2026-10-30","date_precision":"day","short_title":"AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct","deck":null,"takeaway":"The 30 Oct sweep listed 374 arXiv papers that mention AI. We skimmed all their AI statements and read about 35 PDFs. Six results got their own entries: Das–Fox's Murty–Simon proof (GPT-5.6 Sol found the bound), Colbrook–Stepaniants's spectral isometry proof (found by GPT-6 Astra, Lean), the Θ(log k log log k) ratio of greedy for online Steiner forest (GPT Astra, Lean), Gáspár's Laczkovich solution (key proposition by ChatGPT), Ren's pure-semisimplicity counterexample (GPT-6 Astra, Lean) and Petty's conjecture in dimension 5 (Tencent's hyprover agent). PNNL's OpenConjecture study also has one. This listing has the rest, among them a single-author ChatGPT 6 Pro proof of the Alon–Seymour–Thomas separator conjecture, b₄ ≤ 7 for Borsuk's problem and Fiedorowicz's conjecture from a Claude idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":42,"official":42,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic","Google","Tencent","various researchers"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-09-30-barvinok-log-concavity-totally-unimodular-astra","url":"https://postcutoff.com/e/2026-09-30-barvinok-log-concavity-totally-unimodular-astra/","date":"2026-09-30","date_precision":"day","short_title":"GPT-6 Astra proves Barvinok's log-concavity question for contingency tables on lines, giving lattice-point bounds for all totally unimodular polytopes","deck":null,"takeaway":"It is a named question in a central area of combinatorial counting, with the core proof credited to the model.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-30","type":"filed","text":"Created from lead; Liu–Meng abstract and PDF read (science agent)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-30","run":null,"sources_read":"Lead; Liu–Meng abstract and PDF read (science agent)","updated":"2026-10-30","human_review":null,"version":{"date":"2026-10-30"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":172,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":111,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":202,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":141,"in_training_data":false}],"short_url":"https://postcutoff.com/s/counting-perfect-matchings"}