{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra/","md":"https://postcutoff.com/e/2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"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":[{"n":1,"title":"arXiv 2610.12953: Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time","url":"https://arxiv.org/abs/2610.12953","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Research note: conversations between GPT and the first author (chenlin198662/SSP-note)","url":"https://github.com/chenlin198662/SSP-note/blob/main/SSP-research%20note.pdf","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Zhejiang University","OpenAI"],"title":"Subset Sum in O(n + N^(1+ε)) time claimed: Zhejiang team designed the route, GPT Astra Ultra wrote the complete proofs (Chen, Mao, Zhang)","summary":"On 9 Oct 2026 Lin Chen, Yuchen Mao and Guochuan Zhang (Zhejiang University) posted arXiv 2610.12953: Subset Sum on integers of absolute value at most N can be decided deterministically in O_ε(n + N^(1+ε)) time for every fixed ε > 0, and d-dimensional subset sum in O(n + N^(d+ε)), giving a feasibility test for integer programs with d constraints. The paper's \"AI disclaimer\" says the authors designed the main proof route and \"GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs\"; GPT then generalized the one-dimensional randomized algorithm to the deterministic multidimensional one \"without any guidance from the authors\". Unrefereed preprint.","key_facts":["Paper: 'Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time', arXiv 2610.12953 [cs.DS], 9 Oct 2026; Lin Chen, Yuchen Mao, Guochuan Zhang (Zhejiang University); 105 pages","Abstract: 'A long open problem is whether Subset Sum can be solved in O(N) time. We show that Subset Sum can be solved in O(N^(1+ε)) time where ε > 0 is an arbitrarily small constant.'","Corollary 1.2: subset sum for an explicitly listed multiset in [−N, N] 'can be decided deterministically in O_ε(n + N^(1+ε)) word-RAM time, with unrestricted multiplicities'. Theorem 1.1 gives O_{d,ε}(n + N^(d+ε)) for vectors in dimension d","Earlier target-independent bounds in the paper's table: Õ(n + N^(5/3)) (Polak–Rohwedder–Węgrzycki 2021) and Õ(n + N^(3/2)) (Chen–Lian–Mao–Zhang 2024; Jin 2023); Bringmann's 2017 algorithm runs in Õ(n + t), where the target t can be as large as nN","Idea (abstract): dense inputs with the target inside the zonotope were already known to be easy. 'Our key observation is that a sparse input can exhibit a dense structure when lifted to higher dimension'; the thin region near the zonotope's boundary is split into subproblems and solved recursively","AI disclaimer: 'The main proof route for 1D subset sum was designed by the authors … GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs. During the proof, GPT encounters trouble in proving several lemmas, which was eventually resolved by taking ideas from the authors.'","AI disclaimer, continued: 'After obtaining the randomized near-linear time algorithm for 1D subset sum, GPT was able to generalize it directly to deterministic algorithms for multidimensional multiset without any guidance from the authors. GPT was used for drafting the paper. GPT also verified this papers main results using the Lean 4 proof.' A research note with the conversations is linked (github.com/chenlin198662/SSP-note)","Caveats: the Lean proof is described as produced by GPT and is not linked from the abstract page; no independent check or expert reaction was found as of 12 Oct 2026"],"key_numbers":[],"tags":["computer-science","algorithms","subset-sum","integer-programming","additive-combinatorics","gpt-6-astra","lean"],"science":{"field":"computer-science","subfield":"algorithms / fine-grained complexity","problem":"Can Subset Sum be solved in time near-linear in n + N, where N is the largest input integer?","result":"Claimed deterministic O_ε(n + N^(1+ε)) algorithm for Subset Sum and O(n + N^(d+ε)) for d-dimensional subset sum and bounded integer-programming feasibility.","open_since":"","ai_system":["GPT-6 Astra Ultra"],"human_role":"Human-led: the authors designed the proof route and fixed several lemmas; GPT Astra Ultra wrote the complete proofs, generalized the result to the deterministic multidimensional case unaided, drafted the paper and produced a Lean check","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; authors report a Lean 4 verification made by GPT; not independently checked"},"status":"pending","shock":"","short":"Subset Sum in near-linear pseudopolynomial time"},"body_md":"## What happened\n\nSubset Sum asks whether some of n given integers add up to a target. It is NP-complete, but it can be solved in time polynomial\nin the size of the numbers. A long line of work asked how small the dependence on the largest number N can be, with exponents\nfalling from 2 to 5/3 to 3/2. Chen, Mao and Zhang, who wrote several of those earlier papers, now claim an exponent arbitrarily\nclose to 1, which is essentially optimal, and extend it to several dimensions and to integer programs with a fixed number of\nconstraints.\n\nThe paper is explicit about the division of labour. The authors supplied the structure of the argument. GPT Astra Ultra wrote out\nthe proofs, got stuck on several lemmas that the authors resolved, and then generalized the algorithm to the deterministic,\nmultidimensional, multiset case on its own. The conversations are published as a research note.\n\n## Why it matters\n\nIt is a central problem of pseudopolynomial algorithms, claimed by a group with a track record on it. It also shows a working\npattern that is becoming common in theoretical computer science: experts provide the plan, a model produces a very long, detailed\nproof. A 105-page AI-written proof is hard for referees to check, so the claim should be treated as open until others verify it.","disputed":[{"label":"Verification","text":"Unrefereed preprint; authors report a Lean 4 verification made by GPT; not independently checked","sources":[]}],"related":[{"id":"2026-10-08-dualization-eth-lower-bound-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-08-dualization-eth-lower-bound-gpt-6-astra/","date":"2026-10-08","date_precision":"day","short_title":"GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH","deck":null,"takeaway":"Whether monotone Boolean dualization, equivalently enumerating all minimal transversals of a hypergraph, can be done in output-polynomial time has been open since the 1990s. A short preprint says no, assuming the Exponential Time Hypothesis, and its authors say the proof 'was originally discovered by GPT-6 Astra'.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-09","updated":"2026-10-09","orgs":["Hokkaido University","Okayama University","Hosei University","OpenAI"]},{"id":"2026-10-07-colkitt-codex-integer-multiplication-exponent","url":"https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/","date":"2026-10-07","date_precision":"day","short_title":"Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to above 2^-14 (conditional)","deck":null,"takeaway":"In a parallel repository Swapnil Jain (@SJ_Swapnil_Jain, with Claude) posted seven rounds ending at κ ≈ 6.40×10^-5 (> 2^-14) on the evening of 8 Oct, the highest claimed value.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":20,"official":6,"filed":"2026-10-07","updated":"2026-10-08","orgs":["Independent researchers","OpenAI"]},{"id":"2026-10-05-claude-refutes-3sum-apsp-hypotheses","url":"https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/","date":"2026-10-05","date_precision":"day","short_title":"Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses","deck":"Truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)","takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-06","updated":"2026-10-10","orgs":["Anthropic","Columbia University","MIT"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":162,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":101,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":192,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":131,"in_training_data":false}],"short_url":null}