Motivated by applications in road traffic control, we study flows in
networks featuring special characteristics. In contrast to
classical static flow problems, time plays a decisive role.
Firstly, there are transit times on the arcs of the network which
specify the amount of time it takes for flow to travel through a
particular arc; more precisely, flow values on arcs may change over
time. Secondly, the transit time of an arc varies with the current
amount of flow using this arc. Especially the latter feature is
crucial for various real-life applications of flows over time; yet,
it dramatically increases the degree of difficulty of the resulting
optimization problems.
Most problems dealing with flows over time and constant transit
times can be translated to static flow problems in time-expanded
networks. We develop an alternative time-expanded network with
flow-dependent transit times to which the whole algorithmic toolbox
developed for static flows can be applied. Although this approach
does not entirely capture the behavior of flows over time with
flow-dependent transit times, we present approximation results which
provide evidence of its surprising quality.
