On the Scaling Window of Model RB

TL;DR

This paper precisely characterizes the scaling window of the model RB for random constraint satisfaction problems, identifying how the probability of satisfiability transitions around the threshold points.

Contribution

It provides the first analysis of the scaling window for model RB, pinpointing the window's size and behavior near the satisfiability threshold.

Findings

The scaling window width is inversely proportional to n and logarithmic factors.

The satisfiability probability sharply transitions within the derived window.

Results apply to both parameters r and p, with explicit bounds.

Abstract

This paper analyzes the scaling window of a random CSP model (i.e. model RB) for which we can identify the threshold points exactly, denoted by $r_{cr}$ or $p_{cr}$. For this model, we establish the scaling window $W(n,\delta)=(r_{-}(n,\delta), r_{+}(n,\delta))$ such that the probability of a random instance being satisfiable is greater than $1-\delta$ for $r<r_{-}(n,\delta)$ and is less than $\delta$ for $r>r_{+}(n,\delta)$. Specifically, we obtain the following result $$W(n,\delta)=(r_{cr}-\Theta(\frac{1}{n^{1-\epsilon}\ln n}), \ r_{cr}+\Theta(\frac{1}{n\ln n})),$$ where $0\leq\epsilon<1$ is a constant. A similar result with respect to the other parameter $p$ is also obtained. Since the instances generated by model RB have been shown to be hard at the threshold, this is the first attempt, as far as we know, to analyze the scaling window of such a model with hard instances.