A residual-based message passing algorithm for constraint satisfaction problems

TL;DR

This paper introduces a residual-based message passing algorithm that enhances convergence and solution-finding success in complex constraint satisfaction problems, especially near phase transition thresholds.

Contribution

It proposes a novel residual-based updating step for message passing algorithms, improving their performance in solving CSPs around critical thresholds.

Findings

Improved convergence of message passing algorithms.

Higher success probability in finding solutions near thresholds.

Low computational cost of the proposed method.

Abstract

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.