$1.$ Greedy Approximation Algorithm $-$ Trace

The greedy approach selects sets iteratively that cover the most uncovered elements at each step.

Step $1$: Start with all elements uncovered:

$\{1,2,3,4,5,6,7,8\}\backslash \{1,2,3,4,5,6,7,8\backslash \}\{1,2,3,4,5,6,7,8\}$

Choose the set with the most uncovered elements:

A$=\{1,2,3\}$A $=\backslash \{1,2,3\backslash \}$A$=\{1,2,3\}$ covers $3$ elements

B$=\{2,3,4,5\}$B $=\backslash \{2,3,4,5\backslash \}$B$=\{2,3,4,5\}$ covers $4$ elements

C$=\{4,5,6\}$C $=\backslash \{4,5,6\backslash \}$C$=\{4,5,6\}$ covers $3$ elements

D$=\{3,5,7\}$D $=\backslash \{3,5,7\backslash \}$D$=\{3,5,7\}$ covers $3$ elements

E$=\{7,8\}$E $=\backslash \{7,8\backslash \}$E$=\{7,8\}$ covers $2$ elements

Select BBB as it covers the most elements $(4$ elements: $2,3,4,52,3,4,52,3,4,5).$

Covered elements: $\{2,3,4,5\}\backslash \{2,3,4,5\backslash \}\{2,3,4,5\}$

Remaining elements: $\{1,6,7,8\}\backslash \{1,6,7,8\backslash \}\{1,6,7,8\}$

Step $2$: Choose the next set with the most uncovered elements:

A$=\{1,2,3\}$A $=\backslash \{1,2,3\backslash \}$A$=\{1,2,3\}$ adds $1$ new element $(111)$

C$=\{4,5,6\}$C $=\backslash \{4,5,6\backslash \}$C$=\{4,5,6\}$ adds $1$ new element $(666)$

D$=\{3,5,7\}$D $=\backslash \{3,5,7\backslash \}$D$=\{3,5,7\}$ adds $1$ new element $(777)$

E$=\{7,8\}$E $=\backslash \{7,8\backslash \}$E$=\{7,8\}$ adds $2$ new elements $(7,87,87,8)$

Select EEE as it adds the most new elements $(2$ elements: $7,87,87,8).$

Covered elements: $\{2,3,4,5,7,8\}\backslash \{2,3,4,5,7,8\backslash \}\{2,3,4,5,7,8\}$

Remaining elements: $\{1,6\}\backslash \{1,6\backslash \}\{1,6\}$

Step $3$: Choose the next set:

A$=\{1,2,3\}$A $=\backslash \{1,2,3\backslash \}$A$=\{1,2,3\}$ adds $1$ new element $(111)$

C$=\{4,5,6\}$C $=\backslash \{4,5,6\backslash \}$C$=\{4,5,6\}$ adds $1$ new element $(666)$

D$=\{3,5,7\}$D $=\backslash \{3,5,7\backslash \}$D$=\{3,5,7\}$ adds $0$ new elements

Select AAA as it adds $1$ new element $(111).$

Covered elements: $\{1,2,3,4,5,7,8\}\backslash \{1,2,3,4,5,7,8\backslash \}\{1,2,3,4,5,7,8\}$

Remaining elements: $\{6\}\backslash \{6\backslash \}\{6\}$

Step $4$: Choose the next set:

C$=\{4,5,6\}$C $=\backslash \{4,5,6\backslash \}$C$=\{4,5,6\}$ adds $1$ new element $(666)$

Select CCC as it adds the remaining uncovered element $(666).$

Covered elements: $\{1,2,3,4,5,6,7,8\}\backslash \{1,2,3,4,5,6,7,8\backslash \}\{1,2,3,4,5,6,7,8\}$

All elements are now covered.

Greedy Solution

The sets chosen are:

B$,$E$,$A$,$CB$,$ E$,$ A$,$ CB$,$E$,$A$,$C

$2.$ Optimal Solution

To minimize the number of sets, let's analyze:

B$=\{2,3,4,5\}$B $=\backslash \{2,3,4,5\backslash \}$B$=\{2,3,4,5\}$ covers $2,3,4,52,3,4,52,3,4,5$

E$=\{7,8\}$E $=\backslash \{7,8\backslash \}$E$=\{7,8\}$ covers $7,87,87,8$

C$=\{4,5,6\}$C $=\backslash \{4,5,6\backslash \}$C$=\{4,5,6\}$ covers $666$

Combining B$,$C$,$EB$,$ C$,$ EB$,$C$,$E covers all elements:

$\{1,2,3,4,5,6,7,8\}\backslash \{1,2,3,4,5,6,7,8\backslash \}\{1,2,3,4,5,6,7,8\}.$

Thus, the optimal solution is:

B$,$C$,$EB$,$ C$,$ EB$,$C$,$E

Final Answer:

Greedy Solution: B$,$E$,$A$,$CB$,$ E$,$ A$,$ CB$,$E$,$A$,$C

Optimal Solution: B$,$C$,$EB$,$ C$,$ EB$,$C$,$E