Fleischer, Lisa; Skutella, Martin Minimum cost flows over time without intermediate storage. (English) Zbl 1094.90511 Proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, Baltimore, MD, USA, January 12–14, 2003. New York, NY: Association for Computing Machinery; Philadelphia, PA: Society for Industrial and Applied Mathematics (ISBN 0-89871-538-5/pbk). 66-75 (2003). Summary: 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.For the entire collection see [Zbl 1006.00017]. Cited in 1 ReviewCited in 15 Documents MSC: 90B10 Deterministic network models in operations research 68W25 Approximation algorithms Keywords:dynamic flows PDFBibTeX XMLCite \textit{L. Fleischer} and \textit{M. Skutella}, in: Proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, SODA 2003, Baltimore, MD, USA, January 12--14, 2003. New York, NY: Association for Computing Machinery; Philadelphia, PA: Society for Industrial and Applied Mathematics. 66--75 (2003; Zbl 1094.90511)