Computational Methods for Path-based Robust Flows

TL;DR

This paper introduces a computational approach for solving path-based robust flow problems under uncertainty, and proposes a new probabilistic model bridging stochastic and robust optimization, with theoretical and practical validation.

Contribution

It presents a novel computational method for optimal robust flows and introduces a new probabilistic model that balances stochastic and robust optimization, applicable to general linear problems.

Findings

The computational approach effectively solves robust flow problems to optimality.

The new probabilistic model offers a practical compromise between stochastic and robust optimization.

The approach extends to broader linear optimization problems.

Abstract

Real world networks are often subject to severe uncertainties which need to be addressed by any reliable prescriptive model. In the context of the maximum flow problem subject to arc failure, robust models have gained particular attention. For a path-based model, the resulting optimization problem is assumed to be difficult in the literature, yet the complexity status is widely unknown. We present a computational approach to solve the robust flow problem to optimality by simultaneous primal and dual separation, the practical efficacy of which is shown by a computational study. Furthermore, we introduce a novel model of robust flows which provides a compromise between stochastic and robust optimization by assigning probabilities to groups of scenarios. The new model can be solved by the same computational techniques as the robust model. A bound on the generalization error is proven for the case that the probabilities are determined empirically. The suggested model as well as the computational approach extend to linear optimization problems more general than robust flows.

Tables

Table 1. Table 1: Summary of test-cases and the corresponding parameterization.

Network	Parameters	Nodes	Arcs
P1	$M=50,n=20$	3	1920
P2	$n=5,m_0=20$	5	160 (in mean)
NETGEN-a		256	2048
NETGEN-b		512	4096
R-MAT-a	$(a,b,c,d)=(0.5,0.2,0.2,0.1)$	$1000$	$2000$
R-MAT-b	$(a,b,c,d)=(0.5,0.2,0.2,0.1)$	$1000$	$5000$
R-MAT-c	$(a,b,c,d)=(0.2,0.2,0.4,0.2)$	$1000$	$5000$

Table 2. Table 2 : Computational results for various network instances. Maximum of 3 3 3 random samples. Times are 90 % percent 90 90\% quantiles of all pricing/interdiction calls.

$\underset{x\in X}{\max }E\left({\text{Val}}_{\omega }\left(x\right)\right)={\int }_{\Omega }{\text{Val}}_{\omega }\left(x\right)𝑑Q\left(\omega \right).$

(3)

Equations

$$\displaystyle\mathcal{F}=\left\{\left(x_{P}\right)_{P\in\mathcal{P}}\;\Big{|}\;\sum_{\begin{subarray}{c}P\in\mathcal{P}:\\ e\in P\end{subarray}}x_{P}\leq u_{e}\forall e\in E,\;x\geq 0\right\},$$

$$\displaystyle\max\left\{\sum_{P\in\mathcal{P}}x_{P}\;\big{|}\;x\in\mathcal{F}\right\}.$$

$$\displaystyle\max\left\{\sum_{P\in\mathcal{P}}x_{P}-\lambda\;\big{|}\;x\in\mathcal{F},\;\lambda\geq\eta(x)\;\forall\eta\in\binom{E}{k}\right\},$$

$$\min$$

$$\sum _{e\in P}{z}_e+z_P\ge 1$$

$$\sum _{e\in E}{z}_e=k$$

$$z_e\in \{0,1\},z_P\in [0,1]$$

$$\max$$

$$\sum _{e\in {\delta }^{-}(v)}{z}_e-\sum _{e\in {\delta }^{+}(v)}{z}_e=0$$

$$\sum _{e\in {\delta }^{-}(t)}{z}_e-\sum _{e\in {\delta }^{+}(t)}{z}_e=1$$

$$z_e\le z_{\eta }$$

$$z_e,z_{\eta }\in \{0,1\}.$$

$$\max$$

$$\sum _{P:e\in P}{x}_P\le u_e$$

$$\sum _{P:P\cap \eta \ne \varnothing }{x}_P\le \lambda$$

$$x_P\ge 0$$

$$\sum _{P\in \mathcal{P}}{x}_P-k\underset{e}{\max }\sum _{\begin{array}{c}P\in \mathcal{P}:\\ e\in P\end{array}}{x}_P=|x|-k\underset{e}{\max }{x}_e$$

$$\underset{x\in X}{\max }\sum _{k=1}^K{q}_k\underset{\omega \in {\Omega }_k}{\min }{\text{Val}}_{\omega }(x).$$

$$\underset{x\in X}{\max }\sum _{k=1}^K{q}_k\underset{\omega \in {\Omega }_k}{\inf }{\text{Val}}_{\omega }(x)=\underset{Q\equiv q}{\inf }\underset{x}{\max }\int {\text{Val}}_{\omega }(x)dQ(\omega ).$$

$$\underset{Q\equiv q}{\inf }\underset{x\in X}{\max }{E}_Q({\text{Val}}_{\omega }(x))=\underset{x\in X}{\max }\underset{Q\equiv q}{\inf }{E}_Q({\text{Val}}_{\omega }(x))\ge \underset{x\in X}{\max }\sum _{k=1}^K{q}_k\underset{\omega \in {\Omega }_k}{\inf }{\text{Val}}_{\omega }(x).$$

$$E_{Q^n}({\text{Val}}_{\omega }(x))=\sum _{k=1}^K{q}_k{\text{Val}}_{{\omega }_k^n}(x)\to \sum _{k=1}^K{q}_k\underset{\omega \in {\Omega }_k}{\inf }{\text{Val}}_{\omega }(x)\text{, as }n\to \infty .$$

$$\hat{Q}(\eta )=\frac{1}{N}\sum _{i=1}^N{\text{1}}_{\eta ={\eta }_i}=\text{"fraction of days on which exactly arcs in η failed"}.$$

$$\sum _{k=1}^K{q}_k\underset{\eta \in {\Omega }_k}{\min }{\text{Val}}_{\eta }(x)$$

$$-\frac{2}{\sqrt{N}}\sum {\lambda }_k^{\ast }\sqrt{{\hat{q}}_k}-2C\sqrt{\frac{8\ln (4/\delta )}{N}}.$$

$$\sum _{k=1}^K{q}_k\underset{\eta \in {\Omega }_k}{\min }{\text{Val}}_{\eta }(x)-\sum _{k=1}^K{\hat{q}}_k\underset{\eta \in {\Omega }_k}{\min }{\text{Val}}_{\eta }(x)$$

$$=\sum _{k=1}^K\left(\frac{1}{N}\sum _{i=1}^N{X}_i^k-E(X_1^k)\right){\lambda }_k(x)$$

$$=\frac{1}{N}\sum _{i=1}^N\lambda (x{)}^T{X}_i-E_Q(\lambda (x{)}^T{X}_1).$$

$$\frac{1}{N}\sum _{i=1}^N\lambda (x{)}^T{X}_i\ge E_Q(\lambda (x{)}^T{X}_1)-E_Q({\hat{R}}_N)-C\sqrt{\frac{8\ln (4/\delta )}{N}},$$

$${\hat{R}}_N=E_{\sigma }\underset{\lambda :0\le {\lambda }_k\le {\lambda }_k^{\ast }}{\sup }\frac{2}{N}\left|\sum {\sigma }_i({\lambda }^T{X}_i)\right|$$

$$\frac{1}{N}\sum _{i=1}^N\lambda (x{)}^T{X}_i\ge E_Q(\lambda (x{)}^T{X}_1)-{\hat{R}}_N-2C\sqrt{\frac{8\ln (4/\delta )}{N}},$$

$${\hat{R}}_N$$

$$\le E_{\sigma }\frac{2}{N}\sum _{k=1}^K{\lambda }_k^{\ast }\left|\sum _{i:{\eta }_i\in {\Omega }_k}{\sigma }_i\right|$$

$$\le \frac{2}{N}\sum _{k=1}^K{\lambda }_k^{\ast }\sqrt{E_{\sigma }{\left(\sum _{i:{\eta }_i\in {\Omega }_k}{\sigma }_i\right)}^2}.$$

$${\hat{R}}_N$$

Full text

Furthermore, we introduce a novel model of robust flows which provides a compromise between stochastic and robust optimization by assigning probabilities to groups of scenarios. The new model can be solved by the same computational techniques as the robust model. A bound on the generalization error is proven for the case that the probabilities are determined empirically. The suggested model as well as the computational approach extend to linear optimization problems more general than robust flows.

**Keywords: **robust flows, stochastic optimization, robust optimization, generalization error, computational study

1 Introduction

Network flow problems are among the most studied topics in combinatorial optimization, computer science and operations research. In many variants, the complexity status is entirely understood and often very efficient algorithms exist. In this work, we consider a special network flow problem, namely the maximum flow problem, which seeks to maximize the total throughput in a capacitated network. Although it has first been studied by Ford and Fulkerson in the 1950s, new, faster algorithms still appear. The latest contribution of this kind was only four years ago and is due to Orlin, (2013).

The large interest in network flow problems stems from at least two reasons. Network structures appear in a large range of different problems. Thus, a better understanding of the polyhedral structure of network flow problems lead to efficient algorithms for adjoint problems in the past. As an example, the linear relaxation of the precedence constrained knapsack problem can be solved by means of network flow problems very efficiently. This has a direct application to open pit mining (Moreno et al., (2010)). But they have broad, direct applications in practice as well, such as production planning, scheduling, and logistics. Such applications usually are accompanied by data uncertainty. Shipments of parts in a production environment may be delayed, thus, reducing short-term production capacities. Means of transport in a logistics network may be delayed or attacked, thus, resulting in a reduction of the total throughput of particular links. Classical results regarding network flow problems do not take such issues into account but require precise input data. Moreover, they may compute solutions which are optimal for precise data, but may perform bad or turn infeasible after data slightly changed. Hence, the urge for robust solutions is constantly rising.

Network flow problems under uncertainty have been studied in a range of different settings. The structure of the uncertainty set and the demanded type of solution depend on the actual application and influence the problem complexity. Here, we consider the problem in the robust optimization framework as suggested by Bertsimas and Sim, (2003), denoted as the $k$-robust flow problem. Given a graph $G=(V,E)$ with designated vertices $s$ and $t$, we want to find an $s$-$t$-flow which has maximum value after the removal of at most $k\in\mathbb{N}$ edges. In this setting, the flow may not adapt to the set of edges being destroyed but has to be fixed beforehand. Any flow shipped on paths intersecting the set of removed edges is destroyed. The set of possible scenarios can be described as the subsets of edges of cardinality at most $k$.

Adaptive optimization models for problems under uncertainty have also been studied in the past. In such models, the solution may adapt to each of the scenarios individually in order to improve the solution quality. Since the set of scenarios can be of exponential size, the adaptation is usually not provided explicitly. Instead, the solution is accompanied by a recovery algorithm which computes the adaptation on demand. Although such models can provide better solution quality, many practical applications do not support such short-term adaptations. Capacities in production or logistics networks are usually planned and reserved in advance and changes take time in order to be propagated properly. Hence, we consider the robust, more restrictive, model in most of this work.

In practical applications, the worst-case may only appear with a very small probability. In such situations, evaluation of solutions with respect to their worst-case performance may be too pessimistic. Instead, stochastic optimization methods may be applied. In case of maximum flows, one could maximize the expected flow value given a probability distribution for the joint failure of any set of edges. Unfortunately, detailed stochastic models are computationally intractable in most cases. Moreover, specification of the full probability distribution is unreliable since observation of historical events does not necessarily yield good estimates on the probability of events that never appeared before. In Section 5, we consider a model which remedies both aspects by aggregating the occurrence of single events into larger classes.

Related work.

Wood, (1993) showed that the network interdiction problem is NP-hard. The network interdiction problem is related to the robust flow problem with the order of optimization changed. Network interdiction minimizes the maximum flow value of a network, instead of maximizing the minimum flow value in the worst-case. For this problem, Altner et al., (2010) provide valid inequalities and prove integrality gaps for natural LP formulations. Approximability of the network interdiction problem is studied, for example, by Chestnut and Zenklusen, (2017).

Aneja et al., (2001) show how the $k$-robust flow problem (robust flow problem in the following) can be solved for $k=1$. For $k=2$, Du and Chandrasekaran, (2007) showed that the dual separation problem to the $k$-robust flow model we introduce in Section 2 is NP-hard. They reason that - due to the equivalence of optimization and separation - this implies that the optimization problem is also NP-hard. Disser and Matuschke, (2017) pointed out that this reasoning is flawed. Du and Chandrasekaran proved hardness for arbitrary objective functions, whereas the objective function stemming from the robust flow LP is very specific. Hence, the equivalence does not apply and the complexity status remains open, if $k$ is bounded by a constant greater than one. Disser and Matuschke, (2017) showed that the problem is NP-hard, if $k$ is not bounded.

A number of approximation results were shown for robust flow problems. Bertsimas et al., (2013) present an approximation algorithm for $k$-robust flows whose performance may be bounded in terms of the optimal solution. The same algorithm can be shown to achieve the approximation bound $(k+1)/(k^{2}/4+k+1)$. As an alternative approach, Baffier and Suppakitpaisarn, (2014) study $k$-route flows (see also Aggarwal and Orlin, (2002); Kishimoto and Takeuchi, (1992); Kishimoto, (1996)). A $k$-route flow is a conic combination of elementary flows which in turn consist of $k$ disjoint paths. Baffier and Suppakitpaisarn, (2014) showed that a $k$-route flow can be used to obtain a $(k+1)$-approximation for $k$-robust flow and evaluate the practical applicability of their theoretical results (Baffier and Suppakitpaisarn, (2015)). Baffier et al., (2014) also showed that the algorithm sometimes determines an optimum solution to the problem and provide conditions under which this can be checked a posteriori. We note that the approximation bound derived by Bertsimas et al., (2016) is tighter than the latter.

Various other approaches to study network flows subject to arc failures have been suggested in the literature. Matuschke et al., 2017b showed that the robust flow problem becomes tractable if the model is altered such that, instead of edges, specific paths are attacked. In this case, they show that the problem can be solved in polynomial time. They also consider an extension of the model where each arc may be fortified against attacks by paying an arc-specific cost. Aneja et al. discuss so-called $\delta$-reliable flows. A flow is said to be $\delta$-reliable, if no edge carries more than a $\delta$ fraction of the total flow value (Aneja et al., (2007); Baffier et al., (2014)). Such solutions can be computed in polynomial time. Bertsimas et al., (2013) suggest an arc-based flow model which allows for an adaptation once an interdiction has occurred and is closely related to the network interdiction problem. A different adaptation scheme is studied by Matuschke et al., 2017a , who introduce a path-based model which only allows for local adaptations of the scheduled flow.

Gottschalk et al., (2016) discuss the $k$-robust flow problem with an additional temporal component. They provide insight regarding the power of temporally repeated solutions.

Our contribution.

In this paper, we perform a computational study which indicates that, in practice, robust flow problems are tractable in many situations. We show that a natural LP formulation with exponentially many constraints and variables can be solved in reasonable time using simultaneous primal and dual separation techniques. This is partially due to the insight that the robust flow problem admits a sparse solution for various types of networks.

On the theoretical side, we introduce a variant of $k$-robust flow weighting different values of $k$. The new model may be treated by the presented computational approach as well. Furthermore, we state the precise relation of the weighted model to stochastic and robust optimization, generalizing a result of Bertsimas et al., (2016). Our result applies to more general optimization problems under uncertainty than robust flows and presents a compromise between detail and robustness of the description of an uncertain setting. By employing a Rademacher complexity bound, we show that assessing the performance of a flow solution based on historical failure scenarios is more reliable for robust flows than for the solution of a fully stochastic model.

Outline.

The rest of this paper is organized as follows. In Section 2, we introduce our model for robust flows. In Section 3, a separation procedure is described to obtain primal and dual bounds on the robust flow value. Computational results are presented in Section 4. In Section 5, we discuss a connection between a stochastic optimization model for flows under uncertainty and the robust model from Section 2. We introduce a hybrid model which subsumes both problems at its extremes. The paper is concluded with a summary of the results.

2 Robust flow model

Nominal maximum flow problem.

Ford and Fulkerson, (1956) were the first to study the maximum flow problem. They provided the famous and influential max flow - min cut theorem 60 years ago. Although the problem is usually known in its edge based formulation, the first appearance was stated in terms of a path formulation. Given a graph $G=(V,E)$ with edge capacities and two designated vertices $s,t\in V$. Find a set of paths with flow rates such that no edge capacity is exceeded and the total throughput is maximized. Formally, the set of feasible flows can be written as

[TABLE]

where $\mathcal{P}$ denotes the set of all s-t-paths in $G$. By $x_{P}$ we denote the flow rate on path $P\in\mathcal{P}$. The optimization problem is given by

[TABLE]

Nowadays, we know a large variety of algorithms which usually solve the problem in terms of the equivalent edge formulation. While both formulations are equivalent in the nominal case, that is, without any kind of robustness, the actual path decomposition matters when we consider the robust counterpart. See Figure 1 for an example.

Robust maximum flow problem.

In the remainder of this work, we elaborate and evaluate computational techniques which help solving the following robust counterpart of the maximum flow problem. For any feasible flow vector $x\in\mathcal{F}$ we define the flow value with respect to scenario $\eta\in\binom{E}{k}=\{C\subseteq E:|C|=k\}$ as the total amount of flow using paths which avoid edges in $\eta$. That is, $\text{Val}_{\eta}(x)=\sum_{P:P\cap\eta=\emptyset}x_{P}$. Moreover, we denote the robust flow value for $x\in\mathcal{F}$ as the flow value with respect to the worst case scenario, that is, $RVal(x)=\min_{\eta\in\binom{E}{k}}\text{Val}_{\eta}(x)$. The robust maximum flow problem seeks to maximize this value. One way of formulating this problem as a linear program is as follows

[TABLE]

where $\eta(x)=\sum_{P:P\cap\eta\neq\emptyset}x_{P}$ denotes the amount of flow intersecting edges in $\eta$. The formulation ensures that $x$ is a feasible flow. Moreover, the value of $\lambda$ is lower bounded by the amount of flow destroyed by each scenario. Hence, for a feasible solution, $\lambda$ is always at least as large as the amount of flow destroyed in the worst case. Since $\lambda$ has a negative objective function coefficient in a maximization problem, it will also attain this value. Thus, the objective function maximizes the robust flow value.

Aneja et al., (2001) showed how (P) can be solved for $k=1$. Du and Chandrasekaran, (2007) claimed that the problem for every $k\geq 2$ is NP-hard. However, Disser and Matuschke, (2017) recently showed that the proof is incorrect and that the problem is NP-hard if $k$ is not bounded by any constant. The result is not yet published and the problem remains open if $k$ is bounded by a constant. The dual separation problem of (P) remains NP-hard for $k\geq 2$ as shown by Du and Chandrasekaran, (2007). Bertsimas et al., (2016) provide a relaxation of (P) with an approximation ratio of $(k+1)/(k^{2}/4+k+1)$.

3 Solution techniques

Due to its size, model (P) can neither be stored in memory nor solved efficiently. Large scale linear programs with an exponential number of constraints are typically solved by exploiting the specific model structure in order to generate constraints which should be part of the basis. On the other hand, if the number of variables is large, one may generate constraints for the dual LP instead. The latter procedure is known as column generation, as new variables are added to the primal model (Desrosiers and Lübbecke, (2005)).

The robust flow model (P) has both, a large number of variables and constraints. We suggest to solve this issue by generating the set of variables and constraints simultaneously. To this end, we first investigate the primal and dual separation problem individually. Subsequently, we show how both procedures benefit from applying them simultaneously.

Primal separation.

The huge number of constraints of (P) can be tackled by separating them as required. That is, we solve the model for a subset $\mathcal{N}^{\prime}\subseteq\binom{E}{k}$ of constraints and only add scenarios for which the corresponding linear inequality is violated. The separation problem is equivalent to the network interdiction problem, which may be described by the integer program (cf. Bertsimas et al., (2013))

[TABLE]

Problem (1) can be interpreted as an instance of the weighted Maximum Coverage Problem, which is NP-hard in general (see e.g. Ageev and Sviridenko, (1999)). However, the corresponding linear relaxation admits a relative integrality gap which is bounded by $1-e^{-1}$ (Ageev and Sviridenko, (1999)). Hence, one might expect a standard Branch-and-Bound solver to perform reasonably well on this problem, especially if the set $\mathcal{P}$ of paths is of low cardinality. Note that solving (P) for the linearly relaxed separation problem (1) yields the heuristic of Bertsimas et al., (2013).

Dual separation.

In a similar fashion, the large set of variables of (P) can be handled by means of column generation. In each iteration, the model is solved with a subset $\mathcal{P}^{\prime}\subseteq\mathcal{P}$ of path variables to obtain a feasible dual solution $y=(y_{e},y_{\eta})$. The pricing problem, which is the separation problem of the dual LP, is to find a path $P$ which maximizes $\sum_{\eta\cap P\neq\emptyset}y_{\eta}-\sum_{e\in P}y_{e}$. As shown by Du and Chandrasekaran, (2007), this problem is NP-hard in general. Yet, if the vector $(y_{\eta})_{\eta}$ is sufficiently sparse, a solution may be obtained by applying a Branch-and-Bound procedure to the following integer program:

[TABLE]

If the value of (2) is positive, the path encoded by the variables $z_{e}$ is added to the set $\mathcal{P}^{\prime}$ and model (P) is solved again. When solving (2), only those variables $z_{\eta}$ with $y_{\eta}\neq 0$ need to be considered. Thus, sparsity of $y$ simplifies the pricing problem.

Simultaneous separation.

While the primal separation (1) and dual separation (2) problems are both NP-hard, they benefit from a small set of paths and scenarios, respectively. When generating variables and constraints simultaneously, we maintain a set $\mathcal{P}^{\prime}\subseteq\mathcal{P}$ of paths and a set $\mathcal{N}^{\prime}\subseteq\binom{E}{k}$ of scenarios and solve the restricted linear program

[TABLE]

If $\mathcal{P}^{\prime}$ and $\mathcal{N}^{\prime}$ are small, (P’) can be solved fast by a standard LP solver. The dual solution $y$ may be interpreted as setting $y_{\eta}=0$ for all $\eta\notin\mathcal{N}^{\prime}$. Due to this sparsity, the pricing problem (2) may be solved efficiently for the restricted set of scenarios $\mathcal{N}^{\prime}$, generating a new path which is added to the set $\mathcal{P}^{\prime}$ of candidate paths. We keep adding paths until optimality is achieved. At this point, model (P’) with optimal value $f^{d}$ is a relaxation of the full robust flow model (P), and $f^{d}$ is a dual bound on the optimal value $f^{*}$ of the full model.

For this set $\mathcal{P}^{\prime}$ of paths, keep adding scenarios by means of the (restricted) separation problem (1) until it no longer affects the solution $(x,\lambda)$ of model (P’). Then $(x,\lambda)$ is feasible for the full model. Thus, the corresponding value $f^{p}=(\sum_{P}x_{P}-\lambda)$ is a primal bound on $f^{*}$.

Iterating these two steps, the model yields tighter bounds as additional variables and constraints are considered. Once the primal and dual bounds coincide, that is, $f^{p}=f^{d}$, the corresponding flow $f$ is known to be optimal for (P). Since the number of potential variables and constraints is finite, the algorithm eventually terminates. Empirical results indicate that it terminates after a small number of iterations in practice.

Practical aspects.

Once the variables $z_{\eta}$ are fixed, the dual separation problem (2) reduces to a shortest path problem. Thus, if the corresponding price vector $(y_{\eta})_{\eta}$ is very sparse, an optimal path could be determined by enumerating all configurations of $z_{\eta}$ and computing the corresponding shortest path in a reduced network. A similar strategy is realized by only branching on the binary variables $z_{\eta}$, since (2) is totally unimodular for fixed $z_{\eta}$.

When performing the primal and dual separations, it is not necessary to find an optimal solution to the separation problem at each step as long as it yields a violated constraint. Therefore, one could try to tweak the procedure by slightly modifying the separation procedures. In our computational experiments, for the generation of new path variables, we add a small penalty term $\epsilon=10^{-4}$ for each edge which has already been considered by a generated path. For the interdiction problem, we add a penalty of $1/k$ for each arc which is interdicted by a scenario generated previously. If the perturbed problem does not yield a violated constraint, we evoke the unperturbed model. The intuition motivating these modifications is to reduce correlation of the generated constraints.

For very large networks, the required number of primal constraints to solve the primal separation problem could be very large. Numerical experiments reveal that this issue arises in particular for large values of $k$ (see Section 4). As a consequence, each iteration of the simultaneous separation procedure is slowed down, leading to stagnant updates of the primal and dual bounds $\lambda^{p},\lambda^{d}$. As a remedy, we suggest to generate at most a fixed number $c$ of primal constraints in each iteration. Each single call of the interdiction LP (1) yields the robust value of a feasible solution, that is, a primal bound which might improve the current bound. Subsequently, generating paths until optimality for the reduced set of scenarios is achieved restores a dual bound, as described previously. This iteration scheme leads to primal and dual bounds which narrow the optimality gap gradually.

A suitable initialization of the sets $\mathcal{P}^{\prime}$ and $\mathcal{N}^{\prime}$ may help in solving problem (P’). The initial set of paths may be generated by means of the heuristic of Bertsimas et al., (2013). They suggest to find a feasible flow $x\in\mathcal{F}$ which maximizes

[TABLE]

where $x_{e}=\sum_{P\in\mathcal{P}:e\in P}x_{P}$ is the arc flow corresponding to $x$, and $|x|=\sum_{P}x_{P}$. Problem (H) can be solved efficiently using standard maximum flow techniques and binary search. It is shown by Bertsimas et al., (2013) that any path decomposition corresponding to the optimal solution $x^{*}$ of (H) attains the robust value $RVal(x^{*})=|x^{*}|-k\max_{e}x_{e}$. Hence, any such path decomposition is a reasonable initialization for the set $\mathcal{P}^{\prime}$.

4 Computational results

Instances.

The efficacy of the proposed computational decomposition is assessed for several types of networks (see Table 1). Two types of series graphs are considered. The network P1 consists of three nodes $\{s,v,t\}$ with $n$ parallel arcs $(s,v)$, each with (large) capacity $M\in\mathbb{Z}_{+}$, and $(n-1)M$ parallel arcs $(v,t)$ with unit capacity (see Figure 2).

Furthermore, we generate random series graphs of type P2 with nodes $V=\{v_{1},\ldots,v_{n}\}$, $s=v_{1},t=v_{n}$ and parallel arcs $e^{j}_{1},\ldots,e^{j}_{m_{j}}$. The number $m_{j}$ of arcs in the $j$-th cut is randomizes, following a shifted Poisson distribution $m_{j}=m_{0}+\text{Poi}(m_{0})$. Capacities are chosen as $u_{e^{j}_{i}}=u_{e^{j}_{i-1}}+(|\epsilon_{i}^{j}|+1)^{2}$ for independent standard normal random number $\epsilon^{j}_{i}$. Thus, the capacities within a single cut are highly correlated, leading to much variation in the capacity of the cuts.

A third class P3 of series graphs is obtained by connecting a node $s$ to $v$ with a safe arc of capacity $m$ and an interdictable arc of capacity $M$. The node $v$ is connected to $t$ by $m+M$ arcs with capacity $1$. This subgraph is repeated $n$ times to obtain a highly symmetric network of rather small cardinality (see Figure 3).

More complex random networks are generated by a model called NETGEN (Klingman et al., (1974)). These instances consist of multiple sources and sinks which may be cast into the framework of s-t-flows by connecting these to a supersource, respectively supersink, by safe arcs. That is, arcs, which may not be interdicted. We use the NETGEN-8 instances provided as benchmark by the LEMON graph library project (Dezső et al., (2011)).

The R-MAT graph generator (Chakrabarti et al., (2004)) is parameterized by four parameters $(a,b,c,d)$ which lead to different types of clustering within the network. Since R-MAT instances are apriori no flow networks, we choose the node with the largest outdegree as source and the one with the largest indegree as sink. All arcs which are directly connected to the source and sink are safe from interdiction. The arc capacities are chosen independently at random from a uniform distribution on the interval $(0,1)$.

Results.