{"schema":"postcutoff/event@1","as_of":"2026-10-15T23:34:00+02:00","url":"https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/","md":"https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/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":"2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims tight 1 + 2/e approximation for k-median","deck":null,"takeaway":"Family 125 of OpenAI's 722-manuscript release claims a deterministic polynomial-time (1 + 2/e + ε)-approximation for metric k-median, which would settle the problem's approximability under P ≠ NP. OpenAI lists Lean comparator statements for it; no independent expert check has been published yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI math repository: CONTENTS.md (family 125, metric k-median)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI math repository: Lean scope note for family 125","url":"https://github.com/openai/math/blob/main/lean/docs/125.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Lean comparator statement: MetricKMedian.lean","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/MetricKMedian.lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2610.18144: A Better-than-2 Approximation for k-Median Part I (cites OpenAI's result as concurrent work)","url":"https://arxiv.org/abs/2610.18144","type":"paper","group":"primary","domain":"arxiv.org"}],"official":4,"filed":"2026-10-15","updated":"2026-10-15","orgs":["OpenAI"],"title":"OpenAI's math release claims the optimal approximation for metric k-median: a polynomial-time (1 + 2/e + ε) algorithm matching the known hardness, with a Lean formalization","summary":"Among the 719 manuscripts OpenAI published on Oct 6, 2026 (github.com/openai/math), family 125, \"The metric k-median approximation threshold and recovery\" (two manuscripts dated Sept 24, 2026), claims a deterministic polynomial-time (1 + 2/e + ε)-approximation for finite metric k-median with specified candidate facilities, for every fixed ε > 0. Because approximating below 1 + 2/e is NP-hard, this would make 1 + 2/e ≈ 1.736 the exact threshold, where the best published human bound was about 2. OpenAI lists Lean comparator challenges stating the algorithm and the threshold. The EPFL-led better-than-2 paper of Oct 13 cites the claim as concurrent work.","key_facts":["Source: OpenAI's math repository, CONTENTS.md family 125: 'Gives a deterministic polynomial-time (1+2/e+ε)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming P≠NP, the optimal infimum approximation factor is 1+2/e.'","Manuscripts (both dated Sept 24, 2026): 'The approximation threshold for metric k-median' and 'Single-exponential recovery and bounded-price strictness for metric k-median'; the second also gives a randomized (2 − σ)-approximation for an absolute σ > 0","Lean: lean/docs/125.md lists four comparator statements (MetricKMedian, KMedianThreshold, KMedianRecovery, KMedianRefinedRecovery); MetricKMedian.lean states the result with Lean's TM2ComputableInPolyTime (a polynomial-time Turing-machine algorithm), permitted axioms propext, Quot.sound and Classical.choice; the solution module OAI.Combinatorics.KMedian has about 1,300 files (repo tree, Oct 15)","Context: k-median cannot be approximated below 1 + 2/e in polynomial time unless P = NP (Guha–Khuller / Jain–Mahdian–Saberi); before October 2026 the best guarantees were (2 + ε) (Cohen-Addad et al. 2025; Byrka et al. 2026), as summarised in arXiv 2610.18144","Concurrent human work: arXiv 2610.18144 (EPFL, Google Research, CMU, NYU; Oct 13) reaches better than 2 and describes OpenAI's (1 + 2/e + ε) algorithm as extending Byrka et al.'s iterative rounding; a group around Jarosław Byrka and Shi Li independently reached 2 − σ","Production: per the repo README, most results came from an unreleased internal OpenAI model, about three hours of ChatGPT Pro thinking compute per result, from about 4,000 posed problems","Verification status: OpenAI's own Lean formalization; no referee report, expert blog post or independent build check found as of Oct 15"],"key_numbers":[],"tags":["computer-science","algorithms","approximation-algorithms","k-median","lean","openai-math-release"],"science":{"field":"computer-science","subfield":"approximation algorithms","problem":"Exact polynomial-time approximability of metric k-median (gap between the 1 + 2/e hardness and the ~2 upper bound)","result":"Claimed: deterministic polynomial-time (1 + 2/e + ε)-approximation for every fixed ε > 0, matching the NP-hardness bound, so the threshold is exactly 1 + 2/e; plus a randomized (2 − σ)-approximation","open_since":"2002","ai_system":["unreleased internal OpenAI model"],"human_role":"Autonomous per OpenAI: produced by the model in OpenAI's open-problem evaluation; humans curated and released","verification":{"key":"lean","label":"Lean-verified","text":"Lean formalization by OpenAI (comparator statements published); not yet independently checked or refereed"},"status":"pending","shock":"A clean answer to a two-decade-old approximation question in theoretical computer science, appearing as one item among hundreds in a model release."},"body_md":"## What happened\n\nOpenAI's Oct 6, 2026 release of model-written mathematics ([github.com/openai/math](https://github.com/openai/math)) includes a\nfamily of two k-median papers dated Sept 24. The headline claim is a deterministic polynomial-time algorithm that, for any fixed ε > 0,\nfinds k facilities whose total client distance is within (1 + 2/e + ε) of the optimum. Approximating k-median below 1 + 2/e is NP-hard,\nso if correct this settles the problem's approximation ratio in the model with specified candidate facilities. A companion paper gives\na randomized (2 − σ)-approximation through a recovery argument.\n\nOpenAI's Lean catalogue links four \"comparator\" statements for the family. The main one, `MetricKMedian.lean`, states the algorithm's\nexistence and polynomial running time formally in Lean (via Mathlib's Turing-machine complexity definitions). The proof module is large,\nabout 1,300 Lean files.\n\nThe result received little attention in the first week. It surfaced through the EPFL-led paper of Oct 13 (arXiv 2610.18144), whose own\nhuman-proved algorithm breaks factor 2 and which mentions the OpenAI work as concurrent. No independent expert has yet published an\nassessment of the OpenAI proofs or checked the formalization build.\n\n## Why it matters\n\nClosing the gap between the 1 + 2/e lower bound and the factor-2 upper bound for k-median was a well-known open question in\napproximation algorithms. A formalized, model-generated answer would be one of the strongest computer-science items in OpenAI's\nrelease, but it has the same caveats as the rest: self-published, unrefereed, and checked so far only by OpenAI's own Lean work.","disputed":[{"label":"Verification","text":"Lean formalization by OpenAI (comparator statements published); not yet independently checked or refereed","sources":[]}],"related":[{"id":"2026-10-13-k-median-better-than-2-approximation-epfl","url":"https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/","date":"2026-10-13","date_precision":"day","short_title":"k-median breaks the factor-2 barrier","deck":"EPFL-led team gives a 1.95-LMP algorithm; GPT-6 Astra pushes it to 1.844 and proves a 1.8 limit, both with Lean certificates","takeaway":"A human-proved 1.95-LMP algorithm, with a companion paper, gives a better-than-2 approximation for metric k-median. GPT-6 Astra improved the ratio of the same method to 1.844 and proved the method cannot go below 1.8, with Lean certificates. The paper says OpenAI announced other better-than-2 algorithms on 6 Oct.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-15","orgs":["EPFL","Google Research","Carnegie Mellon University","New York University","OpenAI"]},{"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":84,"official":13,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-15","type":"filed","text":"Created from data/leads.md (Lean docs and comparator file read via GitHub)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-15","run":null,"sources_read":"Data/leads.md (Lean docs and comparator file read via GitHub)","updated":"2026-10-15","human_review":null,"version":{"date":"2026-10-15"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-tight-1-2"}