  After the discovery that fixed points of loopy belief propagation coincide
with stationary points of the Bethe free energy, several researchers proposed
provably convergent algorithms to directly minimize the Bethe free energy.
These algorithms were formulated only for non-zero temperature (thus finding
fixed points of the sum-product algorithm) and their possible extension to zero
temperature is not obvious. We present the zero-temperature limit of the
double-loop algorithm by Heskes, which converges a max-product fixed point. The
inner loop of this algorithm is max-sum diffusion. Under certain conditions,
the algorithm combines the complementary advantages of the max-product belief
propagation and max-sum diffusion (LP relaxation): it yields good approximation
of both ground states and max-marginals.
