Rigorous results for random (2+p)-SAT

In recent years there has been signi1cant interest in the study of random k-SAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk(n; m) is formed by selecting uniformly and independently m clauses from Bk and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the “order” of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p ∈ [0; 1], a random (2 + p)-SAT formula, F2+p(n; m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1 − p)m from B2. Using the heuristic “replica method” of statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the behavior of random (2 + p)-SAT formulae. In this paper we give the 1rst rigorous results for random (2 + p)-SAT, including the following surprising fact: for p 6 2=5, with probability 1 − o(1), a random (2 + p)-SAT formula is satis1able i@ its 2-SAT subformula is satis1able. That is, for p 6 2=5, random (2 + p)-SAT behaves like random 2-SAT. c © 2001 Elsevier Science B.V. All rights reserved.