Abstract
The probability P(α,N) that search algorithms for random
satisfiability problems successfully find a solution is studied
as a function of the ratio α of constraints per variable
and the number N of variables. P is shown to be finite if
α lies below an algorithm-dependent threshold αA,
and exponentially small in N above. The critical behaviour is
universal for all algorithms based on the widely used unitary
propagation rule:
P[(1 + 
)αA,N] ∼ exp [ − N1/6 Φ(N1/3)]. Exponents are related to the critical behaviour of
random graphs, and the scaling function Φ is exactly
calculated through a mapping onto a diffusion-and-death problem.