Approximating Highly Satisfiable Random 2-SAT

In this paper we introduce two distributions for the Max-2-SAT problem similar to the uniform distribution of satisfiable CNFs and the planted distribution for the decision version of SAT. In both cases there is a parameter p, 0 <= p <= 1/4d, such that formulas chosen according to both distributions are p-satisfiable, that is, at least (3/4d + p)n clauses can be satisfied. In the planted distribution this happens for a fixed assignment, while for the p-satisfiable distribution formulas are chosen uniformly from the set of all p-satisfiable formulas. Following Coja-Oghlan, Krivelevich, and Vilenchik (2007) we establish a connection between the probabilities of events under the two distributions. Then we consider the case when p is sufficiently large, p = gamma root d log d and gamma > 2 root 2. We present an algorithm that in the case of the planted distribution for any epsilon with high probability finds an assignment satisfying at least (3/4d + p - epsilon)n clauses. For the p-satisfiable distribution for every d there is e(d) (which is a polynomial in d of degree depending on gamma) such that the algorithm with high probability finds an assignment satisfying at least (3/4d+ p - epsilon(d))n clauses. It does not seem this algorithm can be converted into an expected polynomial time algorithm finding a p-satisfying assignment. Also we use the connection between the planted and uniform p-satisfiable distributions to evaluate the number of clauses satisfiable in a random (not p-satisfiable) 2-CNF. We find the expectation of this number, but do not improve the existing concentration results.