Determine whether or not there is a known polynomial$-$time algorithm for solving the problem $($Hamilton Cycle, Travelling salesman problem, Vertex cover, Independent Set, Finding Cliques, Subset Sum, OR Dominating Set$).$

You must justify why there is no known poly$-$time algorithm OR identify a poly$-$time procedure that solves the problem.

$($a$)$ A set of n people attend a political meeting, where ten main issues are to be discussed. Each person attending has created a sublist of issues $($selected from the main set of $10$ issues$)$ that they are most interested in$.$ The organisers would like to select a group of at most k people to come up on stage to head the discussion. The group must be chosen so that each person in the audience has at least one issue in common with at least one person on stage.