Constructing an asympotic phase transition in random binary constraint satisfaction problems

The standard models used to generate random binary constraint satisfaction problems are described. At the problem sizes studied experimentally, a phase transition is seen as the constraint tightness is varied. However, Achlioptas et al. Lecture Notes in computer science, Vol. 1330, 107–120 showed that if the problem size (number of variables) increases while the remaining parameters are kept constant, asymptotically almost all instances are unsatisfiable. In this paper, an alternative scheme for one of the standard models is proposed in which both the number of values in each variable's domain and the average degree of the constraint graph are increased with problem size. It is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satisfiable with probability at least 0.5 and a range in which instances are almost all unsatisfiable; hence there is a crossover point at some value of the constraint tightness between these two ranges. This scheme is compared to a similar scheme due to Xu and Li.