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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
Software:
MAX-2-SAT
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