A New Bound for 3-Satisfiable Maxsat and Its Algorithmic Application

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 2 3 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 2 3 m+ 1 3 mU +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 2 3 m+ k clauses, where k is the parameter.