{"schema":"postcutoff/event@1","as_of":"2026-10-15T10:09:00+02:00","url":"https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/","md":"https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/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":"2026-10-13-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":[{"n":1,"title":"arXiv 2610.18144: A Better-than-2 Approximation for k-Median Part I: The Power of General Bids","url":"https://arxiv.org/abs/2610.18144","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: epfl-thl2/project-k-median (Lean certificates)","url":"https://github.com/epfl-thl2/project-k-median","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["EPFL","Google Research","Carnegie Mellon University","New York University","OpenAI"],"title":"k-median breaks the factor-2 barrier: 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","summary":"arXiv 2610.18144 (13 Oct 2026), \"A Better-than-2 Approximation for k-Median Part I: The Power of General Bids\", by Buhai, Cohen-Addad, Fleischmann, Gupta, Jubés Monforte, Pittu, Stouras, Svensson and Vogl, replaces the linear bid in the Jain–Mahdian–Markakis–Saberi–Vazirani greedy algorithm with general bidding functions. That gives a 1.95-LMP algorithm for facility location, where the factor 2 had essentially stood for two decades, and, with a companion paper, a better-than-2 approximation for k-median. The authors say their proofs were human, but GPT-6 Astra found a bidding function reaching 1.844 and proved that the method cannot go below 1.8, both with Lean certificates.","key_facts":["Paper: arXiv 2610.18144 [cs.DS], 13 Oct 2026; authors at EPFL, Google Research, Carnegie Mellon and NYU (incl. Ola Svensson, Vincent Cohen-Addad, Anupam Gupta)","Main result: a polynomial-time 1.95-LMP (Lagrangian-multiplier-preserving) approximation for facility location using the bid β(α, d) = [α − 1.224 d]+; the previous best LMP factor was 2 − η with η ≈ 2.25 × 10⁻⁷ (Cohen-Addad, Grandoni, Lee, Schwiegelshohn)","k-median: a 1.95-approximation opening k + O(log n) centers; with a (1.89 + ε)-approximation for stable instances in the companion Part II, this gives a better-than-2 polynomial-time approximation. Prior state: (2 + ε) (Cohen-Addad et al. 2025; Byrka et al. 2026); k-median is hard to approximate below 1 + 2/e","GPT-6 Astra (Remark 1.5): 'GPT-6 Astra managed to improve the LMP approximation ratio to 1.844', with a 13-term bidding function, and 'managed to prove that for any admissible bidding function … the supremum of the factor revealing program … has value at least 1.8', so the approach cannot go below 1.8","AI statement: 'The mathematical proofs presented in this work (apart from Remark 1.5) were developed solely by the authors without the use of AI.' GPT-5.6 Sol and GPT-6 Astra were used for proof-checking and writing help; 'The lean certificates mentioned in Remark 1.5 and the certificate for Theorem 4.4 were obtained by GPT-6 Astra'","Lean certificates: github.com/epfl-thl2/project-k-median (reachable on 14 Oct 2026)","Concurrent work, as described in the paper: OpenAI announced on 6 Oct two better-than-2 k-median algorithms, an optimal (1 + 2/e + ε)-approximation extending Byrka et al.'s iterative rounding and a (2 − σ)-approximation; a group around Byrka and Shi Li independently reached 2 − σ. These OpenAI claims were not checked here"],"key_numbers":[],"tags":["computer-science","algorithms","approximation-algorithms","k-median","facility-location","gpt-6-astra","lean"],"science":{"field":"computer-science","subfield":"approximation algorithms","problem":"Is factor 2 a barrier for LMP facility location / metric k-median approximation?","result":"No: a 1.95-LMP algorithm (human proof) and a better-than-2 polynomial-time k-median approximation; GPT-6 Astra improved the LMP factor to 1.844 and proved 1.8 is the limit of the bidding approach (Lean certificates).","open_since":"2002","ai_system":["GPT-6 Astra","GPT-5.6 Sol"],"human_role":"Human-led: main proofs by the authors without AI; GPT-6 Astra contributed the 1.844 improvement, the 1.8 limit and Lean certificates","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; Lean certificates for the AI-found bounds and for Theorem 4.4"},"status":"pending","shock":""},"body_md":"## What happened\n\nThe greedy algorithm of Jain, Mahdian, Markakis, Saberi and Vazirani (2002–03) has been the engine behind the best k-median guarantees through its 2-LMP guarantee for facility location. A 2023 local-search analysis improved it by only about 2.25 × 10⁻⁷. In [arXiv 2610.18144](https://arxiv.org/abs/2610.18144) (13 Oct 2026), a team from EPFL, Google Research, CMU and NYU changes the algorithm itself: clients bid with general functions of their budget and distance instead of the classical linear bid. The bid [α − 1.224 d]+ gives a 1.9499-LMP guarantee, which they make \"log-adaptive\" to get a polynomial-time 1.95-approximation opening k + O(log n) centers. With a (1.89 + ε) algorithm for stable instances from the companion Part II, this gives a better-than-2 approximation for k-median.\n\nThe paper separates human and AI work. The main proofs \"were developed solely by the authors without the use of AI\". In Remark 1.5, GPT-6 Astra found a 13-term bidding function with LMP ratio 1.844, and proved that no admissible bidding function can push this factor-revealing program below 1.8. Lean certificates for both, and for the paper's Theorem 4.4, \"were obtained by GPT-6 Astra\" and are on GitHub.\n\nThe authors also note concurrent work. OpenAI announced on 6 Oct two better-than-2 k-median algorithms, including an optimal (1 + 2/e + ε)-approximation, and a group around Jarosław Byrka and Shi Li independently reached 2 − σ. Those claims were not checked for this entry.\n\n## Why it matters\n\nFactor 2 was a well-known barrier in clustering approximation. Here it falls to human work in the same weeks as OpenAI's claimed AI-found algorithms. The AI's own contribution is typical of the stronger 2026 collaborations: it tuned the construction further and proved where the method stops, with machine-checkable certificates.","disputed":[{"label":"Verification","text":"Unrefereed preprint; Lean certificates for the AI-found bounds and for Theorem 4.4","sources":[]}],"related":[{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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":"2026-10-14","type":"filed","text":"Created from the arXiv paper"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-14","run":null,"sources_read":"The arXiv paper","updated":"2026-10-14","human_review":null,"version":{"date":"2026-10-14"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":166,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":105,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":196,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":135,"in_training_data":false}],"short_url":null}