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The probability P(alpha, N) that search algorithms for random Satisfiabilityproblems successfully find a solution is studied as a function of the ratioalpha of constraints per variable and the number N of variables. P is shown tobe finite if alpha lies below an algorithm--dependent threshold alpha\_A, andexponentially small in N above. The critical behaviour is universal for allalgorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon)alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related tothe critical behaviour of random graphs, and the scaling function Phi isexactly calculated through a mapping onto a diffusion-and-death problem.Comment: 7 pages; 3 figure

Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class