Tight Bounds For Random MAX 2-SAT

For a conjunctive normal form formula F with n variables and m = cn 2-variable clauses (c is called the density), denote by maxF is the maximum number of clauses satisfiable by a single assignment of the variables. For the uniform random formula F with density c = 1 + ε, ε À n−1/3, we prove that maxF is in (1 + ε−Θ(ε3))n with high probability. This improves the known upper bound (1 + ε − Ω(ε3/ ln(1/ε))) due to [6]. The algorithm for the lower bound is also simpler. In addition, we present a simple unified algorithm which not only yields bounds mentioned above for c = 1+ε, but also provides a tight lower bound (3/4c+Θ( √ c))n for large enough c’s. To obtain the bounds for c = 1 + ε, we use the Poisson cloning model and analyze the pure literal algorithm, which is simpler than that of the unit clause algorithm used in [6] (Actually, in [6], it is conjectured that the “pure-literal” rule should give the same result using an alternative analysis.). The Poisson cloning model has been introduced in [13] to simplify analysis of certain algorithms with branching process natures. The model turns out to be almost the same as the uniform model with the same (mean) density. ∗Computer Science Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, E-mail: [email protected]. This work was done when the author visited Microsoft Research. †Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and Department of Mathematics, Yonsei University, Seoul, South Korea, E-mail: [email protected].