Quickest Flows Over Time

Lisa Fleischer, Martin Skutella
<span title="">2007</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
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
more &raquo; ... panded network. We present several approaches for coping with this difficulty. First, 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. Second, 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 time-expanded 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. Key words. network flows, flows over time, dynamic flows, quickest flows, earliest arrival flows, approximation algorithms AMS subject classifications. 90B06, 90B10, 90B20, 90C27, 90C35, 90C59, 68Q25, 68W25 SK 58/4-1 and SK 58/5-3. 1 Earlier work on this topic referred to the problems as dynamic flow problems. Recently the term dynamic has been used in many algorithmic settings to refer to problems with input data that arrives online or changes over time, and the goal of the algorithms described is to modify the current solution quickly to handle the slightly modified input. For the problem of dynamic flows, the input data is available at the start. The solution to the problem involves describing how the optimal flow changes over time. For these reasons, we use the term "flows over time" instead of "dynamic flows" to refer to these problems.
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