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Abstract

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. Solving these problems raises issues that do not arise in standard network flows. One issue is the question of storage of flow at intermediate nodes. In most applications (such as, e.g., traffic routing, evacuation planning, telecommunications etc.), intermediate storage is limited, undesired, or prohibited. The minimum cost flow over time problem is NP-hard. In this paper we 1)~prove that the minimum cost flow over time never requires storage; 2)~provide the first approximation scheme for minimum cost flows over time that does not require storage; 3)~provide the first approximation scheme for minimum cost flows over time that meets hard cost constraints, while approximating only makespan. Our approach is based on a condensed variant of time-expanded networks. It also yields fast approximation schemes with simple solutions for the quickest multicommodity flow problem. Finally, using completely different techniques, we describe a very simple capacity scaling FPAS for the minimum cost flow over time problem when costs are proportional to transit times. The algorithm builds upon our observation about the structure of optimal solutions to this problem: they are universally quickest flows. Again, the FPAS does not use intermediate node storage. In contrast to the preceding algorithms that use a time-expanded network, this FPAS runs directly on the original network.