Question $1$: Consider the Differentiated Set Coveruge Problem:

Input: $n$ items, $U=(1,2,$dots,$n),$ coverage requirements of the itema $f=(f_1,f_2,$dots,$f_n),m$ sets, $s_1,S_2,$dots,$S_{m+1}$ price of the sets $p=(p_1,p_2,$dots,$p_m).$

Let $x=(x_1,x_2,$dots,$x_n)$ be the selection decisions of the sets, $x_{1}in\{0,1\}.$

Output: A minimum price selection $x$ of the sets $S_1,S_2,$dots,$S_a$ that can cover the items is $U$ at least f times.

Ex: Let $n=5$ and $U=(1,2,3,4,5)$ luving coverage requirements $f=(1,2,1,2,1).$

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 $p=(5,6,10,2,4).$

For instance, item $j=2$inU should be covered by at least $f_2=2$ different sets $S_{.}.$ We note that item $2$can be concred by $S_1$ with price $p_1=5,$ by $S_2$ with price $p_2=6$ and by $S_3$ with price $p_3=10.$

Determine a grody selection rule for the sets. Design a greedy algorithen for the Differentiated Set Coverage Problem and report the poeudocode.

Discus the time comploxity of your srecdy atgorithm. Is it efficient? Can your algorithm find the optimum solution?

Implement your algotitim in Python. Uke the inpat given in the example ad report the console outputs and your Python code scripts.