Consider an instance of the Satisfiability Problem, specified by clauses Q over a set of Boolean

variables

$$

$1$

$,$

dots,

$$

$$

$.$

$$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

$$

$$

$,$

$$for some

$$

$,$

$$rather than

$$

$$

$$

$$

$$

$.$

$$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

$$

$1$

$$

$$

$$

$2$

$,$

$$

$1$

$$

$$

$$

$3$

$,$

$$

$2$

$$

$$

$$

$3$

$.$

This is monotone, and indeed the assignment that sets al

$!$

$$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

$$

$1$

$$and

$$

$2$

$$to

$1$

$,$

$$and

$$

$3$

$$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

$$

$,$

$$the problem of Monotone Satisfiability with Few True Variables asks:

Is there a satisfying assignment for the instance in which at most

$$

$$variables are set to

$1$

$?$

$$Show

that HittingSet

$<=$

$$

$$MonotoneSATw

$/$

FewTrueVar

$\text{'}$

s

$\text{'}$

$.$

Hint: Convert an instance of hitting set into CNF

$-$

formula

$$

$$that

Contains as many clauses as sets in the collection of hitting set instance

Is defined over as many variables as there are elements in the universe of hitting set

instance