Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 23 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment τ will satisfy at least 23m+13mU+ρn′ clauses, where m is the number of clauses of F, mU is the number of clauses of F containing a variable from U, n′ is the total number of variables in clauses not containing a variable in U, and ρ is a positive absolute constant. Both U and τ can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MaxSat-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 23m+k clauses, where k is the parameter.
Information and Computation, 2013, Vol 231, Issue October, p. 117-124