Super Solutions of the Model RB

TL;DR

This paper introduces the concept of super solutions in the model RB, establishing a threshold for their expected number using the first moment method, which indicates a phase transition in their existence.

Contribution

It defines and analyzes the threshold for (1,1)-super solutions in the model RB, providing new insights into their robustness and stability.

Findings

Identifies a threshold for the expected number of super solutions

Demonstrates a phase transition in the existence of super solutions

Uses the first moment method to establish the threshold

Abstract

The concept of super solution is a special type of generalized solutions with certain degree of robustness and stability. In this paper we consider the $(1,1)$-super solutions of the model RB. Using the first moment method, we establish a "threshold" such that as the constraint density crosses this value, the expected number of $(1,1)$-super solutions goes from $0$ to infinity.