Delaying satisfiability for random 2SAT

• Authors: Alistair Sinclair;Dan Vilenchik
• Venue: APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
• Year: 2010

Let (C1,C1′), (C2,C2′),..., (Cm,Cm′) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4(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 online choice algorithm in the above process which results 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 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.