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About this paper
Andreas Galanis, Daniel Stefankovic, Eric Vigoda
The authors prove NP-hardness of approximating the Ising partition function, the normalizing sum of weights over 2^N spin assignments. Their dense reduction encodes 3-regular Max-Cut and analyzes gadget biases together through a global potential, without requiring conditional independence. For positive-semidefinite interactions of spectral norm at most 1+N^{-1/2+ε}, approximation within exp(N^{c(ε)}) is NP-hard, near a rapid-mixing boundary but not exponentially hard in N.
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