Belief Propagation Guided Decimation on Random k-XORSAT

TL;DR

This paper analyzes the effectiveness of Belief Propagation Guided Decimation on random k-XORSAT, identifying explicit success thresholds and phase transitions that align with physics predictions and improve prior results.

Contribution

It provides explicit success thresholds for the algorithm on random k-XORSAT and links phase transitions in the decimation process to algorithm performance, confirming and extending physics-based predictions.

Findings

Explicit success threshold for the algorithm's performance.

Identification of phase transitions in the decimation process.

Confirmation of physics predictions relating phase transitions to algorithm success.

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].