Bounding the scaling window of random constraint satisfaction problems

Abstract

The model $$k$$k-CSP is a random CSP model with moderately growing arity $$k$$k of constraints. By incorporating certain linear structure, $$k$$k-CSP is revised to a random linear CSP, named $$k$$k-hyper-$${\mathbb F}$$F-linear CSP. It had been shown theoretically that the two models exhibit exact satisfiability phase transitions when the constraint density $$r$$r is varied accordingly. In this paper, we use finite-size scaling analysis to characterize the threshold behaviors of the two models with finite problem size $$n$$n. A series of experimental studies are carried out to illustrate the scaling window of the model $$k$$k-CSP.