On the Average Similarity Degree between Solutions of Random k-SAT and Random CSPs

TL;DR

This paper introduces the average similarity degree to analyze the structure of solutions in random k-SAT and CSPs, revealing phase transitions and thresholds that relate to problem hardness and algorithm design.

Contribution

It establishes the concept of average similarity degree and proves phase transition phenomena in solution similarity as the ratio of constraints to variables increases.

Findings

Phase transition in average similarity degree for k>4

Threshold points correspond to solution space structure changes

Implications for problem hardness and search algorithm design

Abstract

To study the structure of solutions for random k-SAT and random CSPs, this paper introduces the concept of average similarity degree to characterize how solutions are similar to each other. It is proved that under certain conditions, as r (i.e. the ratio of constraints to variables) increases, the limit of average similarity degree when the number of variables approaches infinity exhibits phase transitions at a threshold point, shifting from a smaller value to a larger value abruptly. For random k-SAT this phenomenon will occur when k>4 . It is further shown that this threshold point is also a singular point with respect to r in the asymptotic estimate of the second moment of the number of solutions. Finally, we discuss how this work is helpful to understand the hardness of solving random instances and a possible application of it to the design of search algorithms.