Consider the problem MAX-TRUE-2SAT, where given a set of n variables {x1, . . . , xn},m clauses C1, . . . , Cm with 2 variables each, and k, one must determine if there exists an assignment where at least k variables are true, and every clause is satisfied. Prove that MAX-TRUE-2SAT is NP-complete

Community Answer

【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/1To prove that MAX-TRUE-2SAT is NP-complete, we need to show two things:MAX-TRUE-2SAT is in NP.MAX-TRUE-2SAT is NP-hard.Let's start with the first part:MAX-TRUE-2SAT is in NP:To show that MAX-TRUE-2SAT is in NP, we need to show that given a solution to a MAX-TRUE-2SAT instance, we can verify it in polynomial time.Suppose we are given an instance of MAX-TRUE-2SAT, consisting of n variables {x1, . . . , xn}, m clauses C1, . . . , Cm with 2 variables each, and k.To verify a solution, we need to check two things:a) At least k variables are true in the assignment.b) Every clause is satisfied in the assignment.We can verify both of these conditions in polynomial time by simply checking each clause and each variable assignment. Thus, MAX-TRUE-2SAT is in NP.Now, we need to show that MAX-TRUE-2SAT is NP-hard:2. MAX-TRUE-2SAT is NP-hard:To show that MAX-TRUE-2SAT is NP-hard, we need to reduce another NP-hard problem to MAX-TRUE-2SAT. Here, we choose the 3SAT problem.The 3SAT problem is the problem of determining whether a given Boolean formula in conjunctive normal form (CNF) with at most 3 literals per clause is satisfiable. We know that 3SAT is NP-complete.Given an instance of 3SAT, with n variables and m clauses, we can construct an instance of MAX-TRUE-2SAT as follows:For each clause in the 3SAT instance, we create two 2SAT clauses as follows:If the clause has three literals (e.g. (x1 ∨ x2 ∨ x3)), we create two 2SAT clauses: (x1 ∨ x2) and (¬x2 ∨ x3). Note that this ensures that at least one of the three literals is true.If the clause has two literals (e.g. (¬x1 ∨ x2)), we create one 2SAT clause: (¬x1 ∨ x2).Thus, we have created a 2SAT instance with 2m clauses, which is satisfiable if and only if the 3SAT instance is satisfiable.Now, suppose the 3SAT instance has a satisfying assignment. We can convert this assignment into a satisfying assignment for the MAX-TRUE-2SAT instance as follows: For each variable in the 3SAT assignment, if it is set to true, we set the corresponding variable in the MAX-TRUE-2SAT assignment to true. If it is set to false, we set the corresponding variable in the MAX-TRUE-2SAT assignment to false. Since each clause in the MAX-TRUE- ... See the full answer