Question $1$: $(50$ pts$)$ Consider the Differentiated Set Coverage Problem: Input: n items, U$=1,2,...,$ n$,$ coverage requirements of the items $=$ f$\_1,$ f$\_2,...,$ f$\_$n$,$ m sets, S$\_1,$ S$\_2,...,$ S$\_$m$,$ price of the sets $=$ p$\_1,$ p$\_2,...,$ p$\_$m$.$ Let $=$ x$\_1,$ x$\_2,...,$ x$\_$m$$ be the selection decisions of the sets, x$\_$i in $\{0,1\}.$ Output: A minimum price selection x of the sets S$\_1,$ S$\_2,...,$ S$\_$m that can cover the items in U at least $$ times. Ex: Let n$=5$ and U$=1,2,3,4,5$ having coverage requirements $=1,2,1,2,1.$ Let m$=5$ and S$\_1=\{1,2\},$ S$\_2=\{2,3,4\},$ S$\_3=\{2,5\},$ S$\_4=\{3,4\},$ S$\_5=\{1,4\}$ with prices $=5,6,10,2,4.$ For instance, item j$=2$ in U should be covered by at least f$\_2=2$ different sets S$\_$i$.$ We note that item $2$ can be covered by S$\_1$ with price p$\_1=5,$ by S$\_2$ with price p$\_2=6$ and by S$\_3$ with price p$\_3=10.1.(20$ 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? $3.(20$ pts$)$ Implement your algorithm in Java. Use the input given in the example and report the console outputs. Report your Java code scripts.