Journal ArticleDOI
Quickest Flows Over Time
Lisa Fleischer,Martin Skutella +1 more
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TLDR
The approach yields fully polynomial-time approximation schemes for the NP-hard quickest min-cost and multicommodity flow problems and shows that storage of flow at intermediate nodes is unnecessary, and the approximation schemes do not use any.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. 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. 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.read more
Citations
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Book ChapterDOI
An Introduction to Network Flows over Time
TL;DR: This chapter gives an introduction into the fascinating area of flows over time—also called “dynamic flows” in the literature—and covers many exciting results that have been obtained over the last fifty years.
Journal ArticleDOI
The Continuous-Time Service Network Design Problem.
TL;DR: An iterative refinement algorithm using partially time-expanded networks that solves continuous-time service network design problems and demonstrates that the algorithm not only solves problems but also obtains an optimal solution at each point in time.
Journal ArticleDOI
Optimization models for large scale network evacuation planning and management: A literature review
TL;DR: This work addresses the late evacuation problem, that means the evacuation of people and eventually critical goods which have stayed at a place endangered by wildfire as long as possible, through its two main combinatorial optimization features.
Journal ArticleDOI
A simplex based algorithm to solve separated continuous linear programs
TL;DR: This work considers the separated continuous linear programming problem with linear data, and presents an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions.
Journal ArticleDOI
Fast, Fair, and Efficient Flows in Networks
TL;DR: It is shown that an s-t-flow that is optimal with respect to the average latency objective is near-optimal for the maximum latency objective, and it is close to being fair.
References
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Book
Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Book
Flows in networks
D. R. Ford,D. R. Fulkerson +1 more
TL;DR: Ford and Fulkerson as mentioned in this paper set the foundation for the study of network flow problems and developed powerful computational tools for solving and analyzing network flow models, and also furthered the understanding of linear programming.
Book
Geometric Algorithms and Combinatorial Optimization
TL;DR: In this article, the Fulkerson Prize was won by the Mathematical Programming Society and the American Mathematical Society for proving polynomial time solvability of problems in convexity theory, geometry, and combinatorial optimization.