Question $1$ : Consider a se$=,...,$ and a collection $1,2,...,$ of subsets of

A $($i$.$e$.,$ for each $).$ We say that a set $$ is a hitting set for the collection

$1,2,...,$

if H contains at least one element from each $,$ that is$,$ if $\backslash$cap $!=$ for each i $($so H "hits" all the sets $).$ We now define the Hitting Set Problem as follows. We are given a set $=,...,,$ a collection $1,2,...,$ of subsets of $,$ and a number $.$ We are asked: Is there a hitting set $$ for $1,2,...,$ such that the size of H is at most $?$ Show that $3<=$

Hint: Design an algorithm so that it converts a $3-$CNF$-$formula $$ with $$ clauses over $$ variables into an instance of Hitting Set with

$.$ The universe that has $2$ elements $($one for each literal in $),$

$.$ The collection that contains exactly $+$ sets. Moreover $$ sets in the collection contain exactly $2$ elements.

$.$ The requested size of the actual hitting set in this instance is $.$

Answer :