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

Christophe Deroulers (1) , Rémi Monasson (1)

1 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha_A, and exponentially small in N above. The critical behaviour is universal for all algorithms 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 to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.

Mots clés: criticality; finite-size scaling; universality; combinatorial search algorithms; random KSAT; computational complexity

Domaines: Mécanique statistique [cond-mat.stat-mech]; Complexité [cs.CC]

Christophe Deroulers, Rémi Monasson. Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class. EPL - Europhysics Letters, 2004, 68, pp.153-159. ⟨10.1209/epl/i2004-10177-6⟩. ⟨hal-00001564v2⟩