Approximating Earliest Arrival Flows in Arbitrary Networks

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 one-third of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is half, which is again tight. We present more results of similar flavor and also show nonapproximability results for finding the best orientation for single and multicommodity maximum flows over time

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“…We are able to show that earliest arrival flows and the approximations developed in [2,9] do not exist in this setting. Theorem 6.…”

Section: Earliest Arrival Flowsmentioning

confidence: 93%

“…For multiple sinks, that is usually not the case if the sinks have a finite demand (see Fig. 10), but approximations are still possible [2,9]. β .…”

Section: Earliest Arrival Flowsmentioning

confidence: 99%

Graph orientation and flows over time

2015

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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%

“…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: 97%

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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“…As our work shows, it may still be possible to get better solutions by focusing on instancessensitive guarantees. It is worth noting that this approach has been pursued, albeit not in a systematic way, in other contexts such as low-distortion embeddings of metrics into geometric spaces, e.g., in [6], subset selection problems [14], and earliest arrival network flows [9]. We hope that our work stimulates further research on instance-sensitive worst-case guarantees for other combinatorial optimization problems.…”

Section: Introductionmentioning

confidence: 97%

Instance-sensitive robustness guarantees for sequencing with unknown packing and covering constraints

Megow

Mestre

2013

Proceedings of the 4th Conference on Innovations in Theoretical Computer Science

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Sequencing problems with an unknown covering or packing constraint appear in various applications, e.g., in real-time computing environments with uncertain run-time availability. A sequence is called α-robust when, for any possible constraint, the maximal or minimal prefix of the sequence that satisfies the constraint is at most a factor α from an optimal packing or covering. It is known that the covering problem always admits a 4-robust solution, and there are instances for which this factor is tight. For the packing variant no such constant robustness factor is possible in general.In this work we address the fact that many problem instances may allow for a much better robustness guarantee than the pathological worst case instances. We aim for more meaningful, instance-sensitive performance guarantees. We present an algorithm that constructs for each instance a solution with a robustness factor arbitrarily close to optimal. This implies nearly optimal solutions for previously studied problems such as the universal knapsack problem and for universal scheduling on an unreliable machine. The crucial ingredient and main result is a nearly exact feasibility test for dual-value sequencing with a given target function. We show that deciding exact feasibility is strongly N P-hard, and thus, our test is best possible, unless P=N P.We hope that the idea of instance-sensitive performance guarantees inspires to revisit other optimization problems and design algorithm tailored to perform well for each individual instance.

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Approximating Earliest Arrival Flows in Arbitrary Networks

Cited by 14 publications

References 15 publications

Graph orientation and flows over time

Graph orientation and flows over time

Graph Orientation and Flows over Time

Instance-sensitive robustness guarantees for sequencing with unknown packing and covering constraints

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