Constraint Satisfaction Problems II

There are two ways in which the SAT problems can be framed: satisfiability problem and optimization problem. The optimization problem asks what is the largest number of clauses we can satisfy. The satisfiability problem is a special case of the optimization problem: it asks whether OPT (φ) = n, where n is the number of clauses in φ. It is obvious that MAX-3SAT is NP-hard since 3SAT itself is NP-complete. 2SAT can be solved in linear time. MAX-2SAT is known to be NP-hard. However, we can attempt to approximate those. We will consider a generalization of MAX-SAT which is called weighted MAX-SAT : each clause will have a weight assigned to it. Each clause ψi has weight wi and the goal is to find such assignment σ that