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Ising spin model is NP-hard with J in {-1,0,1}

October 6, 2012

This is meant to be the high level description of the proof. For more details, I refer the reader to the paper “Computational complexity of Ising spin glass models”.

The reduction starts from the NP-completeness of max-cut in cubic graph. With a bit care, we embed the cubic graph onto a two-level grid and thus proving the NP-hardness of the two-level spin glass.

If we only consider planar graph, then we need to add a magnetic field by reduction from maximum independent set in planar cubic graph. Note here planar graph is not grid, I’m not sure whether 2d grid Ising model with magnetic field is NP-hard or not.

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