  This paper resolves a common complexity issue in the Bethe approximation of
statistical physics and the Belief Propagation (BP) algorithm of artificial
intelligence. The Bethe approximation and the BP algorithm are heuristic
methods for estimating the partition function and marginal probabilities in
graphical models, respectively. The computational complexity of the Bethe
approximation is decided by the number of operations required to solve a set of
non-linear equations, the so-called Bethe equation. Although the BP algorithm
was inspired and developed independently, Yedidia, Freeman and Weiss (2004)
showed that the BP algorithm solves the Bethe equation if it converges
(however, it often does not). This naturally motivates the following question
to understand limitations and empirical successes of the Bethe and BP methods:
is the Bethe equation computationally easy to solve? We present a
message-passing algorithm solving the Bethe equation in a polynomial number of
operations for general binary graphical models of n variables where the maximum
degree in the underlying graph is O(log n). Our algorithm can be used as an
alternative to BP fixing its convergence issue and is the first fully
polynomial-time approximation scheme for the BP fixed-point computation in such
a large class of graphical models, while the approximate fixed-point
computation is known to be (PPAD-)hard in general. We believe that our
technique is of broader interest to understand the computational complexity of
the cavity method in statistical physics.
