A tighter upper bound for random MAX 2-SAT

Given a conjunctive normal form F with n variables and m=cn 2-clauses, it is interesting to study the maximum number maxF of clauses satisfied by all the assignments of the variables (MAX 2-SAT). When c is sufficiently large, the upper bound of f(n,cn)E(maxF) of random MAX 2-SAT had been derived by the first-moment argument. In this paper, we provide a tighter upper bound (3/4)cn+g(c)cn also using the first-moment argument but correcting the error items for f(n,cn), and g(c)=(3/4)cos((1/3)×arccos((4ln2)/c-1))-3/8 when considering the ε3 error item. Furthermore, we extrapolate the region of the validity of the first-moment method is c>2.4094 by analyzing the higher order error items.