{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-23-graph-connectivity-nondeterministic-communication-complexity-chatgpt/","md":"https://postcutoff.com/e/2026-10-23-graph-connectivity-nondeterministic-communication-complexity-chatgpt/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":"2026-10-23-graph-connectivity-nondeterministic-communication-complexity-chatgpt","date":"2026-10-23","date_precision":"day","short_title":"Graph connectivity communication bound settled, with ChatGPT","deck":null,"takeaway":"Zhijun Zhang (INSAIT) proves an optimal Ω(N log N) nondeterministic lower bound for graph connectivity, up from Raz–Spieker's Ω(N log log N) of 1995. He also gives near-optimal randomized and optimal r-round bounds. The paper's AI section says ChatGPT discovered the edge-colouring argument for Lemma 4.6 and produced a weaker first version of the randomized bound.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.32346: Settling Nondeterministic Communication Complexity of Graph Connectivity (Zhijun Zhang)","url":"https://arxiv.org/abs/2610.32346","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-26","updated":"2026-10-26","orgs":["INSAIT","OpenAI"],"title":"Nondeterministic communication complexity of graph connectivity settled at Θ(N log N), open since 1986; ChatGPT found a key lemma's argument","summary":"arXiv 2610.32346 (23 Oct 2026): \"Determining the communication complexity of graph connectivity has been one of the longest standing open problems in communication complexity since the seminal work of Babai, Frankl, and Simon [FOCS'86].\" The paper proves the optimal Ω(N log N) nondeterministic lower bound (previous best Ω(N log log N), Raz–Spieker 1995). It also proves Ω(N·log(min(1/ε, N))) for randomized protocols with error ε, optimal for polynomially small ε, and Ω(N/r·log N) for r-round protocols. The \"AI Methodology\" section says ChatGPT materially affected Lemmas 4.6, 5.3 and C.2.","key_facts":["Previously: the deterministic Θ(N log N) bound is due to Hajnal, Maass and Turán (STOC'88); optimal bounds were known only for 1-round (Sun–Woodruff 2015) and 2-round (Radhakrishnan–Reddy–Venkat, ITCS'26) protocols","Method: bounds the size of 1-monochromatic rectangles via a new problem composing inner product with pointer chasing; extends to other graph properties","AI statement (verbatim, excerpt): 'Extensive interaction with ChatGPT was used to assist with exploring, formulating, and improving proof ideas for several lemmas in this paper. The tool materially affected Lemmas 4.6, 5.3 and C.2. In particular, 1. ChatGPT discovered the edge coloring argument for proving Lemma 4.6; 2. ChatGPT produced a preliminary version of the error-dependent randomized lower bound with worse Ω(log log(1/ε)) dependency on ε … The proof has been significantly simplified and improved by human authors … 3. ChatGPT suggested the use of Hölder’s inequality to optimize the lower order term in proving Lemma C.2.'","Constant-error randomized complexity of connectivity remains open (the paper's bound is optimal only for polynomially small ε)"],"key_numbers":[],"tags":["computer-science","complexity-theory","communication-complexity","chatgpt","ai-assisted-proof","arxiv"],"science":{"field":"computer-science","subfield":"communication complexity","problem":"Communication complexity of graph connectivity (Babai–Frankl–Simon 1986)","result":"Optimal Ω(N log N) nondeterministic lower bound; Ω(N·log(min(1/ε,N))) randomized; Ω(N/r·log N) for r rounds","open_since":"1986","ai_system":["ChatGPT"],"human_role":"Human-led; ChatGPT found the argument for one lemma and a weaker version of another, which the author improved","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nIn the two-party model, Alice and Bob each hold some edges of an N-vertex graph and want to know whether the union is connected.\nSending a spanning forest costs O(N log N) bits, and that was known to be optimal for deterministic protocols since 1988. For\nnondeterministic protocols the best lower bound had been Ω(N log log N) since 1995. Zhijun Zhang closes that gap and nearly\ncloses the randomized case.\n\nHis AI section is specific: ChatGPT \"discovered the edge coloring argument\" for one lemma, gave a weaker first version of the\nerror-dependent randomized bound (which the author then strengthened), and suggested Hölder's inequality for a lower-order term.\n\n## Why it matters\n\nIt settles a problem open for about four decades in a core area of complexity theory, with the AI contribution limited to named\nlemmas. The constant-error randomized case remains open.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-26","type":"filed","text":"Created (arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":"ArXiv PDF read","updated":"2026-10-26","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":176,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":115,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":206,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":145,"in_training_data":false}],"short_url":null}