  Message passing algorithms, whose iterative nature captures well complicated
interactions among interconnected variables in complex systems and extracts
information from the fixed point of iterated messages, provide a powerful
toolkit in tackling hard computational tasks in optimization, inference, and
learning problems. In the context of constraint satisfaction problems (CSPs),
when a control parameter (such as constraint density) is tuned, multiple
threshold phenomena emerge, signaling fundamental structural transitions in
their solution space. Finding solutions around these transition points is
exceedingly challenging for algorithm design, where message passing algorithms
suffer from a large message fluctuation far from convergence. Here we introduce
a residual-based updating step into message passing algorithms, in which
messages varying large between consecutive steps are given high priority in the
updating process. For the specific example of model RB, a typical prototype of
random CSPs with growing domains, we show that our algorithm improves the
convergence of message updating and increases the success probability in
finding solutions around the satisfiability threshold with a low computational
cost. Our approach to message passing algorithms should be of value for
exploring their power in developing algorithms to find ground-state solutions
and understand the detailed structure of solution space of hard optimization
problems.
