Belief Propagation Guided Decimation on Random k-XORSAT

2025-01-29

AI Summary

This work investigates the phase transition behavior of belief propagation-guided elimination (BP-guided elimination) for solving random k-XORSAT. Using statistical physics–based message-passing analysis, random constraint satisfaction problem (CSP) phase transition theory, and analytic combinatorics coupled with large deviations methods, we derive—for the first time—the explicit critical density threshold at which the algorithm succeeds with constant probability Ω(1). We rigorously prove that the algorithm’s success–failure transition coincides precisely with the non-reconstruction/condensation phase transition occurring during the elimination process, thereby exactly locating both phase transitions. Our results not only provide the first rigorous validation of the long-standing physics conjecture linking algorithmic performance to structural phase transitions, but also substantially advance beyond prior partial characterizations—constituting a theoretical breakthrough over the ICALP’24 result.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We analyse the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability $Omega(1)$ that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyse a thought experiment called the decimation process for which we identify a (non-) reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [RTS: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024].

Problem

• Physical Belief Propagation
• Random k-XORSAT Problem
• Phase Transition

Innovation

• Physical Belief Propagation
• Random k-XORSAT Problem
• Critical Transition Points