Problem $3$: Apply $[20$ points$]$

Rural communities commonly have local medical clinics, but they may not have a doctor

assigned to that clinic every day. Instead, a given doctor may be responsible for seeing

patients at multiple clinics on different days throughout the week.

Let$$s assume we have a set of n clinics, and we want to guarantee that each clinic has a

doctor assigned to it at least once a week. We have m potentially available doctors. Each

doctor has a set of clinics that the doctor is willing to visit, if asked, at least once a week.

The question is: for a given number k $<$ m$,$ is it possible to select at most k of the doctors

and be assured that every clinic receives a doctor at least once per week?

As an example, consider a case with $3$ doctors: $1,2,3$ and five clinics: A$,$ B$,$ C$,$ D$,$ E$.$ Doctor

$1$ is willing to visit clinics A$,$ B and C$,$ doctor $2$ is willing to visit A$,$ B$,$ C$,$ and D$,$ and doctor

$3$ is willing to visit clinics B and E$.$ Then for k $=2,$ the answer is yes $($we could choose

either doctors $1$ and $3$ or doctors $2$ and $3),$ but for k $=1,$ the answer is no$.$

Prove that this problem, Clinic Match is an N P$-$Complete problem