On the Quickest Flow Problem in Dynamic Networks – A Parametric Min-Cost Flow Approach

Lin, Maokai; Jaillet, Patrick

doi:10.1137/1.9781611973730.89

Cited by 30 publications

(48 citation statements)

References 11 publications

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“…The cancel-and-tighten algorithm for minimum cost flows [11] runs in strongly polynomial time. By replacing scaling phases of Lin and Jaillet's algorithm with the procedures of the cancel-and-tighten algorithm and by employing the idea of fixed arcs due to Goldberg and Tarjan [11], our result answers the open question, mentioned in [19], of whether we can modify the cost scaling algorithm to produce a strongly polynomial time one. Our algorithm runs in O(nm 2 (log n) 2 ) time, which is better than the only other strongly polynomial time complexity, O(m 2 (log n) 3 (m + n log n)), so far achieved by Burkard et al [4].…”

Section: Introductionmentioning

confidence: 72%

“…Theorem 1 (Lin and Jaillet [19]). The temporally repeated flow derived from a static flow x is a quickest flow if and only if x satisfies the following three conditions.…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…Lin and Jaillet [19] introduced an optimality condition for quickest flows constructed from temporally repeated flows.…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…In this article, we deal with the quickest flow problem. Recently, Lin and Jaillet [19] devised a polynomial time algorithm whose time complexity is the same as that of the cost scaling algorithm of Goldberg and Tarjan [11] for solving the minimum cost (static) flow. Their algorithm employs the cost scaling framework, and it gives a weakly polynomial time complexity, O(nm log(n 2 /m) log(nC)), where n and m are the numbers of nodes and arcs, respectively, and C is the maximum transit time, provided that the transit time is given by an integer value.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of this article is organized as follows: In Section 2, we formulate the quickest flow problem and introduce some notation. We review the optimality condition established by Lin and Jaillet [19] in Section 3. We describe our cancel-andtighten algorithm in Section 4.…”

Section: Introductionmentioning

confidence: 99%

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Cancel-and-tighten algorithm for quickest flow problems

Saho

Shigeno

2016

NETWORKS

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Given a directed graph with a capacity and a transit time for each arc and with single source and single sink nodes, the quickest flow problem is to find the minimum time horizon to send a given amount of flow from the source to the sink. This is one of the fundamental dynamic flow problems. Parametric search is one of the basic approaches to solving the problem. Recently, Lin and Jaillet (SODA, 2015) proposed an algorithm whose time complexity is the same as that of the minimum cost flow algorithm. Their algorithm employs a cost scaling technique, and its time complexity is weakly polynomial time. In this article, we modify their algorithm by adopting a technique to construct a strongly polynomial time algorithm for solving the minimum cost flow problem. The proposed algorithm runs in O(nm 2 (log n) 2 ) time, where n and m are the numbers of nodes and arcs, respectively.

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Section: Introductionmentioning

confidence: 72%

“…Theorem 1 (Lin and Jaillet [19]). The temporally repeated flow derived from a static flow x is a quickest flow if and only if x satisfies the following three conditions.…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…Lin and Jaillet [19] introduced an optimality condition for quickest flows constructed from temporally repeated flows.…”

Section: Optimality Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cancel-and-tighten algorithm for quickest flow problems

Saho

Shigeno

2016

NETWORKS

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The Mixed Evacuation Problem

Hanawa

Higashikawa

Kamiyama

et al. 2016

Combinatorial Optimization and Applications

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A dynamic network introduced by Ford and Fulkerson is a directed graph with capacities and transit times on its arcs. The quickest transshipment problem is one of the most fundamental problems in dynamic networks. In this problem, we are given sources and sinks. Then the goal of this problem is to find a minimum time limit such that we can send the right amount of flow from sources to sinks. In this paper, we introduce a variant of this problem called the mixed evacuation problem. This problem models an emergent situation in which people can evacuate on foot or by car. The goal is to organize such a mixed evacuation so that an efficient evacuation can be achieved. In this paper, we study this problem from the theoretical and practical viewpoints. In the first part, we prove the polynomial-time solvability of this problem in

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An Introduction to Network Flows over Time

Skutella

Research Trends in Combinatorial Optimization

183

151

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Summary.We give an introduction into the fascinating area of flows over timealso called "dynamic flows" in the literature. Starting from the early work of Ford and Fulkerson on maximum flows over time, we cover many exciting results that have been obtained over the last fifty years. One purpose of this chapter is to serve as a possible basis for teaching network flows over time in an advanced course on combinatorial optimization.Flow variation over time is an important feature in network flow problems arising in various applications such as road or air traffic control, production systems, communication networks (e. g., the Internet), and financial flows. In such applications, flow values on arcs are not constant but may change over time. Moreover, there is a second temporal dimension in these applications. Usually, flow does not travel instantaneously through a network but requires a certain amount of time to travel through each arc. In particular, when routing decisions are being made in one part of a network, the effects can be seen in other parts of the network only after a certain time delay. Not only the amount of flow to be transmitted but also the time needed for the transmission plays an essential role.The above mentioned aspects of network flows are not captured by the classic static network flow models. This is where network flows over time come into play. They include a temporal dimension and therefore provide a more realistic modeling tool for numerous real-world applications. Only few textbooks on combinatorial optimization and network flows, however, mention this topic at all; see, e.g., Fulkerson (1962, Chapter III.9) Ahuja et al. (1993, Chapter 19.6) Korte and Vygen (2008, Chapter 9.7) and Schrijver (2003, Chapter 12.5c).The following treatment of the topic has been developed for the purpose of teaching in an advanced course on combinatorial optimization. We concentrate on flows over time (also called "dynamic flows" in the literature) with finite time horizon and constant capacities and constant transit times in a continuous time model. For This work was supported by DFG Research Center MATHEON in Berlin.

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On the Quickest Flow Problem in Dynamic Networks – A Parametric Min-Cost Flow Approach

Cited by 30 publications

References 11 publications

Cancel-and-tighten algorithm for quickest flow problems

Cancel-and-tighten algorithm for quickest flow problems

The Mixed Evacuation Problem

An Introduction to Network Flows over Time

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