Find the maximum (longest) delay to last user in a multicast T. NP complete proof
Refer to Shorted Path Tree (SPT)- https://en.wikipedia.org/wiki/Shortest-path_tree
It ensures that the delay from source node is always minimum for all the nodes including the maximal ones.

Based on your problem description I think it is SPT. If that's the case, it is not a $\text{NP}$-hard problem.