$(2\mathrm{.}5$ points$)$ A faculty committee has $n$ members and is going to vote on $t$ issues during the summer break. On every issue, every professor on the committee can vote "Yes" $"$no$"$ or abstain $($this last one means they do not vote on that issue$).$ Each issue passes if it receives more "yes" votes as it does $"$no$"$ votes $($abstained votes do not count$).$

The Inner Circle problem is this. The input is a table that tells us how every professor on the committee voted on each issue, along with an integer value $k.$ We say a subset $P^{\text{'}}$ of the professors is an inner circle if$,$ for every issue, we can look solely at set $P^{\text{'}}$ to determine the result of the vote each issue passes if and only if it would have passed had only $P^{\text{'}}$ voted on it$.$

Show that the problem of determining if there is an inner circle of size $k$ is $NP-$complete.

Step one is to prove that this problem is in $NP.$ After that, suppose I want to solve vertex cover. I can create an issue for every edge and a professor for each vertex. For each edge, the two professors who are endpoints vote "yes" on that issue and everyone else abstains$.$ An inner circle of size $k$ exactly corresponds to a vertex cover of the same size in the original. A good student response should show that if the original graph has a vertex cover of size $k,$ there is an inner circle of size $k($the professors to whom these $k$ vertices belong are that set$).$