Quickest Flows Over Time

Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. We present several approaches for coping with this difficulty. Firstly, inspired by the work of Ford and Fulkerson on maximal s-t-flows over time (or ‘maximal dynamic s-t-flows’), we show that static, length-bounded flows lead to provably good multicommodity flows over time. Secondly, we investigate ‘condensed’ time-expanded networks which rely on a rougher discretization of time. We prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed timeexpanded network of polynomial size. In particular, our approach yields fully polynomial time approximation schemes for the NP-hard quickest min-cost and multicommodity flow problems. For single commodity problems, we show that storage of flow at intermediate nodes is unnecessary; and our approximation schemes do not use any.