Second moment method for a family of boolean CSP

TL;DR

This paper develops a new approach using the second moment method to establish non-trivial lower bounds for a subclass of boolean CSPs, including monotone 1-in-k-SAT, improving understanding of phase transition thresholds.

Contribution

It introduces a method for computing non-trivial lower bounds for a subclass of boolean CSPs using characteristic solutions and applies it to analyze phase transition thresholds.

Findings

Established bounds for the threshold r_k of monotone 1-in-k-SAT: between log k / k and log^2 k / k.

Provided a new technique for applying the second moment method to specific solutions.

Enhanced understanding of phase transition behavior in boolean CSPs.

Abstract

The estimation of phase transitions in random boolean Constraint Satisfaction Problems (CSP) is based on two fundamental tools: the first and second moment methods. While the first moment method on the number of solutions permits to compute upper bounds on any boolean CSP, the second moment method used for computing lower bounds proves to be more tricky and in most cases gives only the trivial lower bound 0. In this paper, we define a subclass of boolean CSP covering the monotone versions of many known NP-Complete boolean CSPs. We give a method for computing non trivial lower bounds for any member of this subclass. This is achieved thanks to an application of the second moment method to some selected solutions called characteristic solutions that depend on the boolean CSP considered. We apply this method with a finer analysis to establish that the threshold $r_{k}$ (ratio : #constrains/#variables) of monotone 1-in-k-SAT is $\log k/k\leq r_{k}\leq\log^{2}k/k$.