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A tighter upper bound for random MAX $$2$$-SAT. (English) Zbl 1260.68164
Summary: Given a conjunctive normal form $$F$$ with $$n$$ variables and $$m=cn$$ $$2$$-clauses, it is interesting to study the maximum number $$\max F$$ of clauses satisfied by all the assignments of the variables (MAX $$2$$-SAT). When $$c$$ is sufficiently large, the upper bound of $$f(n,cn)=\mathbb{E}(\max F)$$ 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 $$g(c)=(3/4)\cos((1/3)\times\arccos((4\ln 2)/c-1))-3/8$$ when considering the $${\varepsilon}^{3}$$ 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.

##### MSC:
 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity
MAX-2-SAT
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##### References:
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