Long Range Frustration in Finite-Connectivity Spin Glasses: Application to the random K-satisfiability problem

A long range frustration index R is introduced to the finite connectivity Viana–Bray ±J spin glass model as a new order parameter. This order parameter is then applied to the random K-satisfiability (K-SAT) problem to understand its satisfiability transition and to evaluate its global minimum energy. Associated with a jump in R from zero to a finite value, SAT-UNSAT transition in random 3-SAT occurs when the clauses-to-variables ratio α approaches αc(3) = 4.1897. This transition in random 2-SAT occurs at αc(2) = 1, with R remaining to be zero as long as α < 4.459. An accumulation of long range frustration in random 3-SAT of α ≥ αc(3) may explain why it is NP-complete; its absence in random 2-SAT of αc(2) ≤ α < 4.459 suggests that, the maximum of satisfied clauses in such a system may be determined with times scaling polynomially with system sizes. The zero–temperature phase–diagram of the Viana–Bray ±J spin glass model is found to be identical to that of the random 2-SAT. PACS: 89.75.-k, 75.10.Nr, 02.10.Ox