  We examine the dynamic network flow problem under the assumption that the
flow consists of discrete units. The dynamic network flow problem is commonly
addressed in the context of developing evacuation plans, where the flow is
typically treated as a continuous quantity. However, real-world scenarios often
involve moving groups, such as families, as single units. We demonstrate that
solving the dynamic flow problem with this consideration is APX-hard.
Conversely, we present a PTAS for instances where the base graph is a path with
a constant number of nodes. We introduce a `ready time' constraint to the
minsum bin packing problem, meaning certain items cannot be placed in specific
bins, develop a PTAS for this modified problem, and apply our algorithms to the
discrete and dynamic flow problem.
