On the Average Similarity Degree between Solutions of Random k-SAT and Random CSPs
Abstract
Description
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.
22 pages, the final version to appear in Discrete Applied Mathematics
22 pages, the final version to appear in Discrete Applied Mathematics