Delaying Satisfiability for Random 2SAT

Abstract

Let (C ₁,C′₁),(C ₂,C′₂),...,(C _m ,C′ _m ) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all $4\binom{n}{2}$ clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on-line choice algorithm in the above process which results ${\emph{whp}}$ in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well-known fact that a random m-clause formula constructed without the choice of two clauses at each step is unsatisfiable ${\emph{whp}}$ whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k-SAT for any fixed k coincides with the standard satisfiability threshold for random 2k-SAT.