A pair of unit clauses is called conflicting if it is of the form (x), 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 φm clauses, where φ=(√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 φ m+(1-φ)m'+(2-3φ)n''/2 clauses (in F), where n″ is the number of variables in F which are not in F′. We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds m/2 and m(√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√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√5)k variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.
- Fixed parameter tractable
- Lower bound