On April $12025,$ m $>600$ celebrities will visit HKU, each of which is

willing to have a one$-$to$-$one meeting with a HKU student. There are n $>800$ HKU

students interested in meeting with the celebrities, and each has provided at least $3$

but no more than $10$ choices of celebrities. HKU Registry, in arranging the meetings,

has to follow three requirements: $($i$)$ Each student with GPA over $4\mathrm{.}2$ will be arranged

to meet exactly two celebrities; $($ii$)$ each student with GPA over $3\mathrm{.}9$ but less than $4\mathrm{.}2$

will be arranged to meet exactly one celebrity; $($iii$)$ each other student will meet zero

or one celebrity.

$($a$)(30$ points$)$ Feasibility: Give a polynomial$-$time algorithm which takes the student choices and GPA as input and reports whether it is feasible to arrange the

meetings so as to satisfy the three requirements.

$($b$)(20$ points$)$ Optimization: Modify your algorithm in such a way that whenever

it is feasible to satisfy the three requirements, it finds the maximum number of

meetings. What is the time complexity of the algorithm?