{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/","md":"https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/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-integer-programming-linear-exponent-gpt-astra-ultra","date":"2026-10-19","date_precision":"day","short_title":"Integer programming in standard form in N^O(d) time, improving Eisenbrand–Weismantel's N^O(d²): Zhejiang team's ideas, GPT Astra Ultra's proofs (Chen, Mao, Zhang)","deck":null,"takeaway":"Ten days after their Subset Sum claim, the same Zhejiang trio claim to settle a basic question in integer programming. With d constraints and entries bounded by N, the running time exponent drops from O(d²) to linear in d. As before, the authors give the idea and say GPT Astra Ultra wrote the complete proofs.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.25955: Integer Programing: An Algorithm with Exponent Linear in the Dimension","url":"https://arxiv.org/abs/2610.25955","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.12953: Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time (same authors)","url":"https://arxiv.org/abs/2610.12953","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Zhejiang University","OpenAI"],"title":"Integer programming in standard form in N^O(d) time, improving Eisenbrand–Weismantel's N^O(d²): Zhejiang team's ideas, GPT Astra Ultra's proofs (Chen, Mao, Zhang)","summary":"arXiv 2610.25955 (19 Oct 2026), \"Integer Programing: An Algorithm with Exponent Linear in the Dimension\" by Lin Chen, Yuchen Mao and Guochuan Zhang (Zhejiang University), gives a deterministic algorithm for min{cᵀx : Ax = b, ℓ ≤ x ≤ u, x ∈ Zⁿ} with A ∈ Z^(d×n) running in poly(L_in) + f(d)·N^O(d)·(1 + log(c_max + 1))^O(1), where N bounds |A_ij|. The previous best was N^O(d²) (Eisenbrand & Weismantel 2020), and the abstract calls it \"a fundamental open problem whether the exponent O(d²) can be improved to one linear in d\". The AI statement says the authors designed the route and \"GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs\", and that GPT checked the main results in Lean 4. Unrefereed.","key_facts":["Paper: arXiv 2610.25955 [cs.DS], 19 Oct 2026, Lin Chen, Yuchen Mao, Guochuan Zhang (Zhejiang University). Builds on their 9 Oct Subset Sum / IP feasibility paper (arXiv 2610.12953)","Result: deterministic poly(L_in) + f(d) N^O(d) (1 + log(c_max + 1))^O(1) time for standard-form integer programs with d equality constraints and bounded variables; 'Prior to our work, the best known running time was N^O(d²) [Eisenbrand & Weismantel'20] … We answer this question affirmatively'","Technique (contents): feasibility background 'interior is easy, boundary reduces'; LP pricing to 'separate the unavoidable cost from the repair'; an affordable interior/boundary test with a boundary branch ('few large changes, then a smaller problem') and an interior branch","AI statement, verbatim: 'The main proof route was designed by the authors, building upon their recent work on subset sum and integer programming feasibility [11]. The authors came up with the basic idea of adapting the algorithmic technique for integer programming feasibility test to optimization, and GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs. GPT and Claude were used for drafting the paper. GPT also verified this papers main results using the Lean 4 proof.'","Caveats: no Lean repository is linked in the PDF; no independent check or expert reaction found by 20 Oct morning; the companion Subset Sum claim is also unreviewed"],"key_numbers":[],"tags":["computer-science","algorithms","integer-programming","parameterized-complexity","subset-sum","gpt-6-astra","lean"],"science":{"field":"computer-science","subfield":"algorithms / integer programming","problem":"Can integer programs in standard form with d constraints and entries ≤ N be solved in N^O(d) time instead of N^O(d²)?","result":"Claimed deterministic f(d)·N^O(d)·poly-log(c_max) + poly(input) algorithm for bounded standard-form integer programming.","open_since":"","ai_system":["GPT-6 Astra Ultra"],"human_role":"Human-led: the authors designed the proof route and core idea; GPT Astra Ultra wrote the complete proofs and a Lean check","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv preprint; authors report a GPT-made Lean 4 proof that is not linked"},"status":"pending","shock":"","short":"Integer programming with exponent linear in d"},"body_md":"## What happened\n\nEisenbrand and Weismantel's 2020 proximity bound gave integer programs with few constraints an N^O(d²) algorithm. Chen, Mao and Zhang\ncarry over the \"interior is easy, boundary reduces\" approach from their Subset Sum feasibility paper to optimization, and get an\nexponent linear in d. The division of labour matches their earlier paper: the authors supplied the plan, the model wrote the proofs.\n\n## Why it matters\n\nIf it holds, the result changes the best known running time for a core problem of combinatorial optimization. It is also the\nsecond claim in ten days in which one group says a GPT model turned its sketches into complete proofs of a long-standing\nalgorithms question. That puts the weight on independent checking.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint; authors report a GPT-made Lean 4 proof that is not linked","sources":[]}],"related":[{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra","url":"https://postcutoff.com/e/2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra/","date":"2026-10-09","date_precision":"day","short_title":"Subset Sum in O(n + N^(1+ε)) time claimed","deck":"Zhejiang team designed the route, GPT Astra Ultra wrote the complete proofs (Chen, Mao, Zhang)","takeaway":"A long-open question in algorithms, whether Subset Sum can be solved in time near-linear in the largest input number, gets a claimed positive answer. The authors designed the approach; GPT Astra Ultra turned it into a 105-page paper and generalized it without guidance.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Zhejiang University","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"PDF read in full text","updated":"2026-10-20","human_review":null,"version":null},"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/integer-programming-standard"}