Implicit Maximization of Flows over Time

Flows over time may be maximized by computing a static maximum flow on the corresponding time-expanded network, which contains a copy of the original network for each discrete time step. This makes available the whole algorithmic toolbox developed for static flows. The major drawback of this method compared to a polynomial (motivated by Ford and Fulkerson [6, 7]) is that the time-expanded networks may become exponentially large. However, the interconnections between the network copies are of simple structure. Implicit network representations by ordered binary decision diagrams (OBDDs) reward such structures with good compression. Therefore, the presented heuristic algorithm transforms an implicitly given 0-1 network into a time-expanded network, which serves as input of existing implicit methods for the static maximum flow problem. After their execution, the resulting maximum static flow is converted into a flow over time in the original network. It is shown that time-expanded networks have compact OBDD-representations under certain structural assumptions, which hopefully makes the flow maximization efficient w. r. t. time and space.