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Quadratic-Time Hardness of Furthest Pair in Superconstant Dimension

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furthest-pair-seth-hardnessTheoretical computer scienceposed by fine-grained complexity literaturerecorded: solved

1 attempt · no person has looked

Statement

Furthest Pair and its relatives admit f(d) n2−Θ(1/d)f(d)\,n^{2-\Theta(1/d)} algorithms, making them the standard examples of barely subquadratic computation, and whether that is optimal in superconstant dimension was open. Under SETH it is: Furthest Pair requires quadratic time once the dimension is superconstant.

Context

The barely-subquadratic class is a well-known frontier in fine-grained complexity, and this pins the dimension at which the speedup dies.

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  • #1

    Attempt 1

    proof attemptChatGPT 5.5 Pro (with Codex, Claude Opus, Gemini for feedback) with Barna Saha, Yinzhan Xu, Christopher Ye ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT 5.5 Pro (with Codex, Claude Opus, Gemini for feedback)
    people
    Barna Saha, Yinzhan Xu, Christopher Ye

    The paper states the proof was initially discovered by ChatGPT 5.5 Pro, that the initial prompt was essential, and that other systems were used to generate feedback on it. The authors validated and substantially edited the proof to improve it.

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