A residual-based message passing algorithm for constraint satisfaction problems
This work addresses a specific bottleneck in algorithm design for hard optimization problems, offering an incremental improvement for researchers in computational complexity and constraint satisfaction.
The paper tackled the challenge of message passing algorithms failing to converge near satisfiability thresholds in constraint satisfaction problems by introducing a residual-based updating step that prioritizes messages with large fluctuations. The result was improved convergence and increased success probability in finding solutions around the threshold for the model RB prototype, with low computational cost.
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.