$($b$)$ Imagine you are managing a team of employees, each with their own set of tasks to

complete. However, some tasks caniot be worked on simultaneously due to

dependencies or resource constraints. Your goal is to maximize productivity by

scheduling as many tasks as possible while ensuring that no two conflicting tasks are

assigned to the same team member simultaneously. You described the problem to your

friend. He commented, "The problem can be mapped to a popular NP$-$complete

problem and can be proved to be NP$-$hard by reduction technique."

Justify your friend's comment by mapping the problem to an NP$-$complete problem

and reducing it to another known NP$-$complete problem.

$($c$)$ Recall the deletion operation of the Fibonacci Heap data structure. Professor Rahsut

has proposed variant of the deletion operation $($Figure for Q $5(c))$