We first present an empirical study of the Belief Propagation (BP) algorithm,
when run on the random field Ising model defined on random regular graphs in
the zero temperature limit. We introduce the notion of maximal solutions for
the BP equations and we use them to fix a fraction of spins in their ground
state configuration. At the phase transition point the fraction of
unconstrained spins percolates and their number diverges with the system size.
This in turn makes the associated optimization problem highly non trivial in
the critical region. Using the bounds on the BP messages provided by the
maximal solutions we design a new and very easy to implement BP scheme which is
able to output a large number of stable fixed points. On one side this new
algorithm is able to provide the minimum energy configuration with high
probability in a competitive time. On the other side we found that the number
of fixed points of the BP algorithm grows with the system size in the critical
region. This unexpected feature poses new relevant questions on the physics of
this class of models.Comment: 20 pages, 8 figure
