A tighter upper bound for random MAX 2-SAT

Authors:
Xuelin Xu;Zongsheng Gao;Ke Xu
Affiliations:
LMIB & School of Mathematics and Systems Science, Beihang University, Beijing, 100191, PR China;LMIB & School of Mathematics and Systems Science, Beihang University, Beijing, 100191, PR China;State Key Laboratory of Software Development Environment, Beihang University, Beijing, 100191, PR China
Venue:
Information Processing Letters
Year:
2011

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Abstract

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)xarccos((4ln2)/c-1))-3/8 when considering the @e^3 error item. Furthermore, we extrapolate the region of the validity of the first-moment method is c2.4094 by analyzing the higher order error items.