{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean/","md":"https://postcutoff.com/e/2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean/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":"2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean","date":"2026-10-29","date_precision":"day","short_title":"Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra","deck":null,"takeaway":"In online Steiner forest, terminal pairs arrive one by one and must be connected at once. Awerbuch, Azar and Bartal proved in 1996 that the greedy algorithm is O(log² k)-competitive and conjectured O(log k). On 29 Oct a University of Maryland group led by MohammadTaghi Hajiaghayi posted matching bounds of Θ(log k log log k). If correct, that refutes the conjecture and separates greedy from the best online algorithm by a log log k factor. Some proofs 'were developed with the help of OpenAI's GPT Astra', with Gemini Argon assisting, and Lean files for the main theorem are on GitHub. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.40569: The Tight Competitive Ratio of Greedy for Online Steiner Forest","url":"https://arxiv.org/abs/2610.40569","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean formalization (igholami/online-steiner-forest-lean)","url":"https://github.com/igholami/online-steiner-forest-lean","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Google DeepMind","University of Maryland"],"title":"Maryland team claims greedy for online Steiner forest is Θ(log k · log log k)-competitive, against Awerbuch–Azar–Bartal's 1996 O(log k) conjecture; GPT Astra and Gemini Argon helped, Lean files public","summary":"arXiv 2610.40569 (29 Oct 2026), \"The Tight Competitive Ratio of Greedy for Online Steiner Forest\" by Iman Gholami, Ilya Hajiaghayi, MohammadTaghi Hajiaghayi and Peyman Jabbarzade (University of Maryland; Ilya Hajiaghayi listed at Takoma Park Middle School), proves that the greedy algorithm for online Steiner forest is O(log k log log k)-competitive on all instances and gives a matching Ω(log k log log k) lower bound. This answers the 1996 question of Awerbuch, Azar and Bartal negatively: greedy is not O(log k). GPT Astra helped develop some proofs, Gemini Argon assisted in parts, and the main theorem is formalised in Lean.","key_facts":["Result (paper, verbatim): 'we resolve the competitive ratio of the greedy algorithm for online Steiner Forest, proving that it is tightly Θ(log k log log k) … Our results refute the O(log k) conjecture of Awerbuch, Azar, and Bartal and establish a tight Θ(log log k) separation between the greedy algorithm and the best possible online algorithm for Steiner Forest'","Prior state (abstract): Awerbuch–Azar–Bartal (SODA 1996) proved O(log² k) and conjectured O(log k); Bamas, Drygala and Maggiori (SODA 2022) got O(log k log log k) for greedy on a restricted class of inputs. The best online algorithm is O(log k)","AI statement (verbatim): 'Some of the proofs in this paper were developed with the help of OpenAI's GPT Astra, which also contributed to the writing and exposition, with Gemini Argon assisting in some parts. The authors are responsible for the correctness of the results and the final content of the paper.'","Formal proof: 'Lean files formalizing the main theorem are available on GitHub at https://github.com/igholami/online-steiner-forest-lean'","People: MohammadTaghi Hajiaghayi (University of Maryland) is a senior researcher in network design and online algorithms; Gholami and Jabbarzade are also at Maryland; co-author Ilya Hajiaghayi's affiliation is given as Takoma Park Middle School","Verification status: preprint with a public Lean development (not checked by us); not refereed; no HN thread found on 30 Oct"],"key_numbers":[],"tags":["computer-science","algorithms","online-algorithms","network-design","competitive-analysis","gpt-6-astra","gemini-4-argon","lean","preprint"],"science":{"field":"computer-science","subfield":"online algorithms and network design","problem":"Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal (1996) conjectured O(log k)","result":"Claimed tight bound Θ(log k log log k), so the O(log k) conjecture is false","open_since":"1996","ai_system":["GPT-6 Astra","Gemini 4 Argon"],"human_role":"AI-assisted: GPT Astra helped develop some proofs and the write-up, Gemini Argon assisted in parts; the authors take responsibility","verification":{"key":"lean","label":"Lean-verified","text":"Preprint with public Lean files for the main theorem; not refereed"},"status":"pending","shock":""},"body_md":"## What happened\n\nSteiner forest asks for the cheapest set of edges connecting given pairs of terminals. In the online version the pairs arrive one at a\ntime, and the obvious greedy rule connects each new pair by a shortest path, treating bought edges as free. Since 1996 greedy was known\nto be within O(log² k) of optimal, and its authors conjectured O(log k), which the best online algorithm achieves.\n\nThe Maryland group shows greedy's true ratio is Θ(log k log log k): an upper bound for all instances and a construction where greedy does\nthat badly. The paper credits GPT Astra with help on some proofs and the writing, and Gemini Argon in parts. The main theorem is\nformalised in Lean in a public repository.\n\n## Why it matters\n\nIt closes a 30-year-old question in online algorithms, and with a negative answer: greedy is slightly worse than conjectured. It is also\na TCS result where two labs' models are named and the theorem comes with a formal proof.","disputed":[{"label":"Verification","text":"Preprint with public Lean files for the main theorem; not refereed","sources":[]}],"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"]}],"people":[{"id":"ilya-sutskever","name":"Ilya Sutskever","url":"https://postcutoff.com/person/ilya-sutskever/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-30","type":"filed","text":"Created (science agent; arXiv abstract and PDF read, AI statement and Lean link quoted)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-30","run":null,"sources_read":"Science agent; arXiv abstract and PDF read, AI statement and Lean link quoted","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":182,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":121,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":212,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":151,"in_training_data":false}],"short_url":"https://postcutoff.com/s/online-steiner-forest-greedy"}