Robust flows over time: models and complexity results

Gottschalk, Corinna; Koster, Arie M. C. A.; Liers, Frauke; Peis, Britta; Schmand, Daniel; Wierz, Andreas

doi:10.1007/s10107-017-1170-3

Cited by 7 publications

(19 citation statements)

References 23 publications

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“…Extensions seek to determine a set of quickest flows (Chin-Chia & Yih-Wenn, 2017;Fleischer & Skutella, 2002;Fleischer & Skutella, 2007;Fleischer & Tardos, 1998;Köhler & Skutella, 2005;Melchiori & Sgalambro, 2018;Miller-Hooks & Patterson, 2004;Saho & Shigeno, 2017) that minimize the total time to transmit flows from a source to a sink. More recent literature considers robust flows over time that also consider transit flow times (Gottschalk et al, 2018).…”

Section: Dynamic Flowsmentioning

confidence: 99%

Minimum‐cost flow problems having arc‐activation costs

Curry

Smith

2021

Naval Research Logistics

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Time-dependent network applications, such as wireless sensor network and infrastructure optimization settings, may require dynamic flows to be transmitted according to a nonsimultaneous schedule of path-flows. We study a dynamic network flow optimization problem considering the presence of activation costs required to begin transmitting flow on an arc. This problem can be modeled as a dynamic version of the minimum-cost flow problem having arc-activation costs (MCFA). The MCFA is related but not equivalent to the fixed-charge network flow problem. We first discuss the relationship between these two problems, and show how MCFA is unique in the network flow literature. We present a mixed-integer programming (MIP) model along with a series of symmetry-breaking inequalities for solving the MCFA. As an alternative, we employ a relaxation-based algorithm that iteratively obtains upper and lower bounds via the solution of a series of smaller, more tractable MIPs. We show that this algorithm finitely terminates with an optimal MCFA solution. Finally, computational results demonstrate the efficacy of our approach compared to solving a MIP using a state-of-the-art commercial solver.

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Section: Dynamic Flowsmentioning

confidence: 99%

Minimum‐cost flow problems having arc‐activation costs

Curry

Smith

2021

Naval Research Logistics

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“…Matuschke et al [8] investigated variants of robust flows in which an adversary can target individual flow paths and the network can be fortified against such attacks. Gottschalk et al [9] devised a robust variant of flows over time in which transit times are uncertain.…”

Section: Problem Definitionmentioning

confidence: 99%

The complexity of computing a robust flow

Disser

Matuschke

2020

Operations Research Letters

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Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure. One of the most basic models is the Maximum Robust Flow problem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i.e., the guaranteed value of surviving flow after removal of any k arcs in the network. The complexity of this problem appeared to have been settled a decade ago: Aneja et al. [1] showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran [2] established that the problem is NP -hard for any constant value of k larger than 1.We point to a flaw in the proof of the latter result, leaving the complexity for constant k open once again. For the case that k is not bounded by a constant, we present a new hardness proof, establishing NP -hardness even for instances where the number of paths is polynomial in the size of the network. We further show that computing optimal integral solutions is already NP -hard for k = 2 (whereas for k = 1, an efficient algorithm is known) and give a positive result for the case that capacities are in {1, 2}.

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“…In [5], an adaptive robust model is introduced for the case that arcs might fail. In [15], one instead assumes that at most Γ arcs are delayed. Clearly, if this delay is chosen to be greater than the time horizon, this corresponds to arc failure.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, if this delay is chosen to be greater than the time horizon, this corresponds to arc failure. The model in [15], which we call dynamic path model (DPM), is path-based and can be seen as a generalization of (PM) to flows over time. [15] prove several results on the complexity of (DPM) and they also show that temporally repeated flows, a solution concept in the nominal case, are not optimal in the robust setting but satisfy certain approximation guarantees.…”

Section: Introductionmentioning

confidence: 99%

“…The model in [15], which we call dynamic path model (DPM), is path-based and can be seen as a generalization of (PM) to flows over time. [15] prove several results on the complexity of (DPM) and they also show that temporally repeated flows, a solution concept in the nominal case, are not optimal in the robust setting but satisfy certain approximation guarantees. In the dynamic setting, the path model (DPM) is quite restrictive due to the strict flow conservation and capacity constraints for any possible delay and therefore yields very conservative solutions as mentioned in [15].…”

Section: Introductionmentioning

confidence: 99%

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Robust static and dynamic maximum flows

Biefel¹,

Martina²,

Liers³

et al. 2022

Preprint

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We study the robust maximum flow problem and the robust maximum flow over time problem where a given number of arcs Γ may fail or may be delayed. Two prominent models have been introduced for these problems: either one assigns flow to arcs fulfilling weak flow conservation in any scenario, or one assigns flow to paths where an arc failure or delay affects a whole path. We provide a unifying framework by presenting novel general models, in which we assign flow to subpaths. These models contain the known models as special cases and unify their advantages in order to obtain less conservative robust solutions.We give a thorough analysis with respect to complexity of the general models. In particular, we show that the general models are essentially NP-hard, whereas, e.g. in the static case with Γ = 1 an optimal solution can be computed in polynomial time. Further, we answer the open question about the complexity of the dynamic path model for Γ = 1. We also compare the solution quality of the different models. In detail, we show that the general models have better robust optimal values than the known models and we prove bounds on these gaps.

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Robust flows over time: models and complexity results

Cited by 7 publications

References 23 publications

Minimum‐cost flow problems having arc‐activation costs

Minimum‐cost flow problems having arc‐activation costs

The complexity of computing a robust flow

Robust static and dynamic maximum flows

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