Consider an instance of the Satisfiability Problem, specified by clauses C_1, \ldots, C_k over a set of Boolean variables x_1, \ldots, x_n. We say that the instance is monotone if each term in each clause consists of a nonnegated variable; that is, each term is equal to x_i, for some i, rather than \bar{x}_i. Monotone instances of Satisfiability are very easy to solve: They are always satisfiable, by setting each variable equal to 1 . For example, suppose we have the three clauses
\left(x_1 \vee x_2\right),\left(x_1 \vee x_3\right),\left(x_2 \vee x_3\right) .
This is monotone, and indeed the assignment that sets all three variables to 1 satisfies all the clauses. But we can observe that this is not the only satisfying assignment; we could also have set x_1 and x_2 to 1 , and x_3 to 0 . Indeed, for any monotone instance, it is natural to ask how few variables we need to set to 1 in order to satisfy it.

Given a monotone instance of Satisfiability, together with a number k, the problem of Monotone Satisfiability with Few True Variables asks: Is there a satisfying assignment for the instance in which at most k variables are set to 1?

Prove this problem is NP-complete. Hint: reduce from vertex cover.