Consider an instance of the Satisfiability problem, specified by clauses C1,...,Ck over a set of Boolean variables x1,...,xn. We say that the instance is monotone if each term in each clause consists of a non-negated variable; that is, each term is equal to xi, for some i, rather than xi.
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
(x1 ∨ x2), (x1 ∨ x3), (x2 ∨ x3).
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 x1 and x2 to 1, and x3 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