{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-04-huffman-unequal-letter-costs-np-complete-codex/","md":"https://postcutoff.com/e/2026-11-04-huffman-unequal-letter-costs-np-complete-codex/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":"2026-11-04-huffman-unequal-letter-costs-np-complete-codex","date":"2026-11-04","date_precision":"day","short_title":"Huffman coding with unequal letter costs is NP-complete (Codex)","deck":null,"takeaway":"L. Sunil Chandran and Rishikesh Gajjala (arXiv 2611.04444, 3 Nov 2026) prove that deciding whether a binary prefix-free code of cost at most K exists is NP-complete when the two letter costs are part of the input, even for costs (M, M+1).","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.04444: Binary Huffman Coding with Unequal Letter Costs Is NP-Complete","url":"https://arxiv.org/abs/2611.04444","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Indian Institute of Science"],"title":"Optimal prefix codes with unequal letter costs are NP-hard: Chandran and Gajjala settle a 65-year-old question with a reduction developed by OpenAI's Codex","summary":"L. Sunil Chandran and Rishikesh Gajjala (arXiv 2611.04444, 3 Nov 2026) prove that deciding whether a binary prefix-free code of cost at most K exists is NP-complete when the two letter costs are part of the input, even for costs (M, M+1). The question had been open since Karp's 1961 integer-programming formulation. The paper's AI table says OpenAI's Codex was used for the approach, proof development, writing and checks, but not for asking the question. Unrefereed preprint.","key_facts":["Theorem 1.1: deciding whether there is a binary prefix code of cost at most K is NP-complete, even when (c0, c1) = (M, M+1) with M part of the input; all numbers encoded in binary","Context: Huffman (1952) solved equal letter costs; Karp (1961) gave an integer program; Golin–Rote (1998) and Bradford–Golin–Larmore–Rytter (2002) gave polynomial algorithms for every fixed pair of integer costs; Golin–Mathieu–Young (SICOMP 2012) a PTAS","Proof: two polynomial-time reductions from Planar Independent Set through a problem on full ordered binary trees with prescribed leaf depths","AI table: 'Asking the research question No · Coming up with the approach Yes · Proof development Yes · Writing and exposition Yes · Checking for bugs Yes'","Disclosure: 'The authors initially asked OpenAI's Codex for an NP-hardness proof for the generalized prefix-coding problem … Codex contributed to the reduction strategy and its mathematical development … The authors guided the development, requested corrections and simplifications'","Verification: preprint, no formal proof; no Hacker News thread or expert comment found on 4 Nov"],"key_numbers":[],"tags":["math","computer-science","complexity","information-theory","codex","openai","arxiv"],"science":{"field":"computer-science","subfield":"computational complexity / coding theory","problem":"Complexity of optimal prefix-free coding with unequal letter costs","result":"NP-complete even for letter costs (M, M+1) given in the input","open_since":"1961","ai_system":["Codex"],"human_role":"AI-assisted: authors posed the question and guided; Codex produced the reduction strategy and much of the proof","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nOn 3 Nov 2026 (arXiv listing of 4 Nov) Chandran and Gajjala posted a proof that optimal binary prefix coding with letter costs supplied in the input is NP-hard. The abstract calls it \"this longstanding open problem\" and the title the resolution of a \"65-year-old\" question. Polynomial algorithms exist for each fixed pair of integer costs, so hardness needs the costs to grow with the input.\n\nThe paper's \"AI Methodology\" section has a table: the question came from the authors; the approach, proof development, writing and bug checking used Codex. \"Codex contributed to the reduction strategy and its mathematical development, including the local tree constructions, their assembly into a tree representation of an independent-set computation, and the conversion to unequal-letter-cost coding.\"\n\n## Why it matters\n\nA classical problem in information theory, listed as open in surveys of prefix coding for decades, appears settled with the key reduction attributed to a coding agent. It is an unrefereed preprint; no specialist (e.g. Golin) has commented publicly yet.","disputed":[{"label":"Verification","text":"Unrefereed preprint","sources":[]}],"related":[{"id":"2026-11-04-ai-assisted-results-november-4-listing","url":"https://postcutoff.com/e/2026-11-04-ai-assisted-results-november-4-listing/","date":"2026-11-04","date_precision":"day","short_title":"AI-assisted results, 4 Nov: arXiv of 4 Nov and Hexagon of 2–3 Nov","deck":null,"takeaway":"The 4 Nov sweep listed 374 arXiv papers that mention AI (193 new), 104 conjecture papers and 33 uncited Hexagon works. Three got own entries: Chandran–Gajjala's NP-completeness of Huffman coding with unequal letter costs (Codex), Yang's (1+1/e)Δ colouring mixing claim (GPT-6 Astra Pro) and Akhmedova–Lam–Yost-Wolff's ChatGPT-found counterexample to the amplituhedron magic number conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":60,"official":60,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Anthropic","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-04","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-04","run":null,"sources_read":"The arXiv PDF","updated":"2026-11-04","human_review":null,"version":{"date":"2026-11-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":188,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":127,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":218,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":157,"in_training_data":false}],"short_url":"https://postcutoff.com/s/huffman-coding-unequal-letter"}