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Kemeny Rank Aggregation for Three Voters

Theoretical computer science · posed by Cynthia Dwork, Ravi Kumar, Moni Naor & D. Sivakumar, 2001 · solved

2 comments

Statement

Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even n≥4n \ge 4; three voters was the minimal open case, and n=2n = 2 is polynomial-time solvable.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    GPT-5.6 Sol Ultra, Claude Fable 5, with Dominik Peters

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    GPT-5.6 Sol Ultra found the reduction from MAX CUT; Claude Fable 5 helped simplify parts of it. Together with earlier results, every fixed number of voters n≥3n \ge 3 is now hard.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    The reduction is Lean-checked, alongside an author-written arXiv preprint.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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