Approximating Earliest Arrival Flows with Flow-Dependent Transit Times

Baumann, Nadine; Köhler, Ekkehard

doi:10.1007/978-3-540-28629-5_46

Cited by 8 publications

(14 citation statements)

References 12 publications

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“…Notice that p refers to the earliest arrival pattern of the undirected network in this case. We are able to show that earliest arrival flows and the approximations developed in [1,7] do not exist in this setting.…”

Section: Price In Terms Of Flow and Time Horizonmentioning

confidence: 93%

See 1 more Smart Citation

Graph Orientation and Flows over Time

Arulselvan

Groß

Skutella

2014

Algorithms and Computation

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Abstract. Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems -streets, tracks, etc. -are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send 1 3 of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is 1 2 which is again tight. We present more results of similar flavor and also show non-approximability results for finding the best orientation for single and multicommodity maximum flows over time.

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Section: Price In Terms Of Flow and Time Horizonmentioning

confidence: 93%

“…Earliest arrival flows always exist if only one sink is present, as was first proven by Gale [6]. For multiple sinks, that is usually not the case, but approximations are still possible [1,7]. For every time θ ∈ R + , let f * θ be a maximum flow over time with time horizon θ.…”

Section: Price In Terms Of Flow and Time Horizonmentioning

confidence: 96%

Graph Orientation and Flows over Time

Arulselvan

Groß

Skutella

2014

Algorithms and Computation

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“…Gale (1959) showed the existence of earliest arrival flows for general networks with constant transit times on the arcs and, more generally, for networks with timedependent (but not flow-dependent) transit times and capacities on the arcs. In case of flow-dependent transit times there exists an s-t flow over time that sends the maximum amount of flow from s to t for any time horizon T. But there may not be such a maximum s-t flow that is maximal for each θ [0, T ) (Baumann & Kӧhler 2007). We will show this by the following simple example.…”

Section: Non-existence Of Earliest Arrival Flowsmentioning

confidence: 98%

Flow Over Time Problem With Inflow-Dependent Transit Times

Khanal

Pyakurel

Dhamala

2018

J. Inst. Sci. Tech.

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Network flow over time is an important area for the researcher relating to the traffic assignment problem. Transmission times of the vehicles directly depend on the number of vehicles entering the road. Flow over time with fixed transit times can be solved by using classical (static) flow algorithms in a corresponding time expanded network which is not exactly applicable for flow over time with inflow dependent transit times. In this paper we discuss the time expanded graph for inflow-dependent transit times and non-existence of earliest arrival flow on it. Flow over time with inflow-dependent transit times are turned to inflow-preserving flow by pushing the flow from slower arc to the fast flow carrying arc. We gave an example to show that time horizon of quickest flow in bow graph GB was strictly smaller than time horizon of any inflow-preserving flow over time in GB satisfying the same demand. The relaxation in the modified bow graph turns the problem into the linear programming problem.

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“…Baumann and KÃűhler [7,8] show that for the case of flow-dependent transit times earliest arrival flows do not always exist. However, an optimization version of the earliest arrival problem is introduced.…”

Section: Models and Algorithmsmentioning

confidence: 99%