Problem $8.$ Assume you have an efficient algorithm to solve the maximum flow problem in a given flow network. Use this algorithm to reduce and solve the following problems:

a$).$ Minimum Vertex$-$Disjoint Paths in a Directed Acyclic Graph $($DAG$)$: Given a directed acyclic graph $\backslash ($ G$=($V$,$ E$)\backslash ),$ determine the minimum number of vertex$-$disjoint paths needed to cover all vertices in $\backslash ($ G $\backslash ).$

b$).$ Cycle Cover in a Directed Graph: Given a directed graph $\backslash ($ G$=($V$,$ E$)\backslash ),$ find a cycle cover, which is a collection of vertex$-$disjoint cycles that cover every vertex in $\backslash ($ G $\backslash ),$ or correctly report that no cycle cover exists.

Reduce the problems to a network flow problems. Construct the corresponding flow network and explain how the solution to the maximum flow problem determines the answer.