a$)$ What is the meaning of the concepts P$,$ NP$,$ NP$-$complete, NP$-$hard.

b$)$ If it is known that the independent set problem is NP$-$complete, show that the vertex cover problem is also NP$-$complete.

c$)$ Is there a polynomial algorithm for the vertex cover problem when the graph is a tree? If YES give the algorithm. If NO explain why.

d$)$ Does the vertex cover problem have a good $/$ fast $2-$approximation algorithm that is at most twice as large as an optimal one? If YES write an algorithm that finds an approximate solution. If NO explain why.