Robust and Adaptive Network Flows

Abstract

We study network flow problems in an uncertain environment from the viewpoint of robust optimization. In contrast to previous work, we consider the case that the network parameters (e.g., capacities) are known and deterministic, but the network structure (e.g., nodes and arcs) is subject to uncertainty. In this paper, we study the robust and adaptive versions of the maximum flow problem and minimum cut problems in networks with node and arc failures, and establish structural and computational results. The adaptive two-stage model adjusts the solution after the realization of the failures in the network. This leads to a more flexible model and yields less conservative solutions compared to the robust model. We show that the robust maximum flow problem can be solved in polynomial time, but the robust minimum cut problem is NP-hard. We also prove that the adaptive versions are NP-hard. We further characterize the adaptive model as a twoperson zero-sum game and prove the existence of an equilibrium in such games. Moreover, we consider a path-based formulation of flows in contrast to the more commonly used arc-based version of flows. This leads to a different model of robustness for maximum flows. We analyze this problem as well and develop a simple linear optimization model to obtain approximate solutions. Furthermore, we introduce the concept of adaptive maximum flows over time in networks with transit times on the arcs. Unlike the deterministic case, we show that this problem is NP-hard on series-parallel graphs even for the case that only one arc is allowed to fail. Finally, we propose heuristics based on linear optimization models that exhibit strong computational performance for large-scale instances.

Protection of flows under targeted attacks

“…However, as soon as the interdictor is allowed to remove two arcs, the corresponding dual separation problem becomes NP -hard as was shown by Du and Chandrasekaran [11]. On the positive side, Bertsimas et al [7] building upon a generalization of the parametric LP used in [2], gave an LP-based approximation, in terms of the amount of flow removed by the interdictor in an optimal solution, for the case where the interdictor can remove any given number of arcs B I . More recently, Bertsimas et al [6] showed that the same flow also yields a 1 + (B I /2) 2 /(B I + 1)-approximation.…”

“…More recently, Bertsimas et al [6] showed that the same flow also yields a 1 + (B I /2) 2 /(B I + 1)-approximation. Formally, the model considered in [2,7,11] is defined for the uncertainty set Ω = {z ∈ {0, 1} A | 1 T z ≤ B I } and the flow player aims at solving max x∈X min z∈Ω val(x, z), where val(x, z) = P ∈P (1 − max e∈P z e )x P is the amount of flow x ∈ X that survives after the interdictor selects the scenario z ∈ Ω. A fractional version of this uncertainty set was proposed in [9]: Each arc has a cost c e for destroying the entire capacity on e and the interdictor is allowed to fractionally attack the capacities on the arcs, i. e., Ω = {z ∈ [0, 1] A | c T z ≤ B I }.…”

“…We are given a directed graph D = (V, A) with source s ∈ V and sink t ∈ V , and arc capacities u ∈ R A + . We consider flows on paths, as in [7,Section 4], and so let P denote the collection of all s-t-paths in D. The strategy choices of the decision maker (whom we call the flow player ) are given by the set…”

“…This tractability of the interdictor's optimal strategy is a desirable property of our model as it allows computation of the robust value of any given flow. Note that, in contrast, for the models in [7,17] discussed below, the interdictor's optimal answer to a given flow is NP -hard to compute in general (in both cases, the interdictor's problem is equivalent to the budgeted maximum coverage problem [7]). Also, with the exception of the basic model in [2], no efficient algorithm or constant factor approximation is known for the flow player's problem in the robust flow models discussed below despite intense research.…”

The complexity of computing a robust flow

“…Several alternative robustness models for flows have been proposed in different application contexts. Taking a less conservative approach, Bertsimas, Nasrabadi, and Stiller [3] and Matuschke, McCormick, and Oriolo [7] proposed different models of flows that can be rerouted after failures occur. 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.…”