A pair of unit clauses is called conflicting if it is of the form (x), $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28:411–421, 1981) showed that for each UCF CNF formula with m clauses we can simultaneously satisfy at least $\hat{ \varphi } m$ clauses, where $\hat{ \varphi }=(\sqrt{5}-1)/2$. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula F with m clauses we can find, in polynomial time, a subformula F′ with m′ clauses such that we can simultaneously satisfy at least $\hat{ \varphi } m+(1-\hat{ \varphi })m'+(2-3\hat {\varphi })n''/2$ clauses (in F), where n″ is the number of variables in F which are not in F′.#R##N##R##N#We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds m/2 and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31:335–354, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most 6k+3 variables and 10k clauses. We improve this to 4k variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.
['Robert Crowston', 'Gregory Gutin', 'Mark Jones', 'Anders Yeo']
A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Applications