Question $1$: $(50$ pts$)$ Consider the Differentiated Set Coverage Problem: Input: n items, U $=(1,2.....$n$),$ coverage requirements of the items f $=(1\mathrm{.}12....$ S$.),$ m sets, S$.,$ Sz$.....$ S$.$m$.$ price of the sets p $=($ PP$...$ Pm$)$ Let x $=(11,12,...$ Im$)$ be the selection decisions of the sets, $11(0,1).$ Output: A minimum price selection x of the sets Si$,$ Sa$,...,$ S$.$m that can cover the items in U at least f times. Ex: Let n $=5$ and U $=(1\mathrm{.}2\mathrm{.}3\mathrm{.}4\mathrm{.}5)$ having coverage requirements f $=(1,2,1,2,1).$ Let m $=5$ and S $=\{1,2\},$ S$2=\{2,3,4),$ S$.=\{2,5),$ Ss $=\{3,4\},$ Ss $=\{1,4\}$ with prices p$=(5,6,10,2\mathrm{.}4).$ For instance, item ; $=2$ U should be covered by at least $12=2$ different sets S$.$ We note that item $2$ can be covered by S$,$ with price pi $=5,$ by S$,$ with price p$=6$ and by S$3$ with priceps $=10.1.(20$ pts$)$ Determine a greedy selection rule for the sets. Design a greedy algorithm for the Differentiated Set Coverage Problem and report the pseudocode. $2.(10$ pts$)$ Discuss the time complexity of your greedy algorithm. Is it efficient? Can your algorithm find the optimum solution?