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Approximate Counting for Spin Systems on Planar Graphs

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planar-spin-systems-approximationTheoretical computer scienceposer not recordedrecorded: solved

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Statement

Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting qq-colourings on planar graphs is NP-hard for every constant q≥4q \geq 4, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.

Context

Settles the planar case of computational phase transitions in approximate counting, a well-worked programme in theoretical computer science, though not a single named conjecture.

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Sol Ultra with Heng Guo, Xinyuan Zhang ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol Ultra
    people
    Heng Guo, Xinyuan Zhang

    "The main ideas of all proofs in this paper were found by GPT-5.6 Sol Ultra. For consistency with standard mathematical exposition, the words we and our are used throughout the paper, including when presenting ideas that originate in the output of the model. The authors simplified, streamlined, and wrote all of the proofs." The paper singles out finding the right problem to reduce from as where the model was particularly helpful.

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