# General Bounds on Satisfiability Thresholds for Random CSPs via Fourier Analysis

## TL;DR

This paper introduces general bounds for the satisfiability thresholds of random boolean CSPs using Fourier analysis, applicable to any fixed constraint function, and demonstrates their effectiveness through theoretical and experimental validation.

## Contribution

It provides the first fully general bounds on CSP thresholds based on Fourier spectra, applicable to any constraint function, advancing understanding of phase transitions in random CSPs.

## Key findings

- Bounds are within a constant factor of the true thresholds for many CSPs.
- Bounds can be computed from the Fourier spectrum of constraints.
- Experimental results validate the bounds' effectiveness.

## Abstract

Random constraint satisfaction problems (CSPs) have been widely studied both in AI and complexity theory. Empirically and theoretically, many random CSPs have been shown to exhibit a phase transition. As the ratio of constraints to variables passes certain thresholds, they transition from being almost certainly satisfiable to unsatisfiable. The exact location of this threshold has been thoroughly investigated, but only for certain common classes of constraints. In this paper, we present new bounds for the location of these thresholds in boolean CSPs. Our main contribution is that our bounds are fully general, and apply to any fixed constraint function that could be used to generate an ensemble of random CSPs. These bounds rely on a novel Fourier analysis and can be easily computed from the Fourier spectrum of a constraint function. Our bounds are within a constant factor of the exact threshold location for many well-studied random CSPs. We demonstrate that our bounds can be easily instantiated to obtain thresholds for many constraint functions that had not been previously studied, and evaluate them experimentally.