$1.$ Suppose problem A is in class NP$,$ problem B is in class P$.$ If we can reduce A to B and B to A $($in polynomial time$)$ what do we know? Justify your answer.

$2.$ Outline an algorithm that can quickly find a maximal independent set, on par with how nearest neighbor finds a quick path for TSP$.$ It doesn$$t need to be the maximum set, but it should be a good candidate.

$3.$ Consider what NP$-$complete means in the context of computational complexity theory. Provide a definition and explain why solving an NP$-$complete problem efficiently would have significant implications.

$4.$ Argue correctness of Floyd Warshall algorithm, ie show it considers every viable path.

$5.$ Give the function for retracing the Floyd$-$Warshall path table. The path table helps navigate the shortest path from u to v$.$ Pseudo code is acceptable.