Interdiction Problem Research Articles

An arc interdiction vehicle routing problem with information asymmetry

Abstract In some cases, due to security, sensitivity or high value, it is possible that shipments are interrupted by some adversary decision makers. The vehicle routing problem (VRP) for these kinds of shipments is some more complicated in comparison with classical VRPs, and it has been considered recently by researchers. In reality, two decision makers may have different perceptions or asymmetric information about the network. We study vehicle routing network interdiction problem with information asymmetry and investigate the benefits and risks of considering this assumption. By taking the average of benefits and risks over some network instances, it is observed that when more budgets is assigned to interdiction and the distributor estimates the interdictor's parameters with less accuracy, Network Interdiction Vehicle Routing Problem (NIVRP) with information asymmetry is able to formulate the real cases, more effectively.

Stochastic shortest path network interdiction with a case study of Arizona–Mexico border

One of the key challenges in securing the U.S.-Mexico border is the smuggling of illicit goods and humans between Ports-of-Entry (POEs). A confluence of factors advantageous to traffickers including inconsistent levels of fencing, favorable terrain, and expansive knowledge of specific pathways have contributed to the establishment of preferred routes of illicit transit, yet little is known about the strategic interaction between adversaries and defenders between the POEs. To address this challenge, this paper studies a stochastic shortest-path network interdiction problem where the attacker (drug smugglers, illegal immigrants, or terrorists) attempts to minimize the expected shortest traveling time between the source and the destination, while the defender attempts to maximize the attacker’s expected shortest traveling time by allocating sensors to the arcs to detect the attacker with a limited budget. Using a probabilistic detection likelihood, we formulate bi-level max-min mixed-integer problems on a multi-modal licit and illicit transportation network along the Arizona–Mexico border considering single source and single destination, and multiple sources and multiple destinations, respectively. We find that (a) the expected shortest traveling time will increase as the budget/detection probability increase; (b) the expected shortest time by walking is more than 3 times long than by driving; (c) the multiple sources and multiple destinations model which allows the attacker to choose a random source-destination pair leads to a shorter time than the single source and single destination model. A graphical user interface (GUI) is developed to assist decision making and demonstrate the results.

A defender-attacker model and algorithm for maximizing weighted expected hitting time with application to conservation planning

ABSTRACTThis article studies an interdiction problem in which two agents with opposed interests, a defender and an attacker, interact in a system governed by an absorbing discrete-time Markov chain. The defender protects a subset of transient states, whereas the attacker targets a subset of the unprotected states. By changing some of the transition probabilities related to the attacked states, the attacker seeks to minimize the Weighted Expected Hitting Time (WEHT). The defender seeks to maximize the attacker’s minimum possible objective, mitigating the worst-case WEHT. Many applications can be represented by this problem; this article focuses on conservation planning. We present a defender–attacker model and algorithm for maximizing the minimum WEHT. As WEHT is not generally a convex function of the attacker’s decisions, we examine large-scale integer programming formulations and first-order approximation methods for its solution. We also develop an algorithm for solving the defender’s problem via mixed-integer programming methods augmented by supervalid inequalities. The efficacy of the proposed solution methods is then evaluated using data from a conservation case study, along with an array of randomly generated instances.

A tri-level r -interdiction median model for a facility location problem under imminent attack

Abstract When systems lose their critical linkage and key facilities due to natural disasters or intentional attacks, their performance may be at risk. This paper focuses on reducing the effect of these events, in which facilities can be located and fortified with a limited budget. To address this issue, a conceptual framework is proposed for contributing to strategic decisions-making by considering the worst possible condition in interdiction problems. Hence, we apply a tri-level facility location r-interdiction median (TFLRIM) model based on leader-follower games to minimize the total cost before and after interdiction. Beside, locating the facilities is carried out based on the worst case scenario when an attacker disrupts the system under uncertainty. To cope with uncertain parameters in the presented model, the Me method is utilized. We propose four hybrid meta-heuristics based on three algorithms, namely Tabu Search (TS), Rainfall Optimization (RFO) and Random Greedy Search (RGS). These algorithms are employed to solve 24 random instances, in which their performances are evaluated based on the comparison of their solutions with the obtained exact solutions by the explicit enumeration method. The results show that the hybrid algorithm of RFO and RGS outperforms the current algorithms. Finally, sensitivity analyses of the model shows that proper design of defensive systems has an effective role in reducing the losses of such these systems.

Stochastic maximum flow interdiction problems under heterogeneous risk preferences

We consider a generic maximum flow interdiction problem that involves a leader and a follower who take actions in sequence. Given an interdiction budget, the leader destroys a subset of arcs to minimize the follower’s maximum flows from a source to a sink node. The effect from an interdiction action taken on each arc is random, following a given success rate of decreasing the arc’s capacity to zero. The follower can add additional arc capacities for mitigating flow losses, after knowing the leader’s interdiction plan but before realizing the uncertainty. We consider risk-neutral and risk-averse behaviors of the two players and investigate five bi-level/tri-level programming models for different risk-preference combinations. The models incorporate the expectation, left-tail, and right-tail Conditional Value-at-Risk (CVaR) as commonly used convex risk measures for evaluating random maximum flows in the leader’s and follower’s objectives. We reformulate each model as an equivalent mixed-integer linear program and test them on real-world network instances to demonstrate interactions between the leader and the follower under various risk-preference settings.

Exact algorithms and bounds for the dynamic assignment interdiction problem

AbstractWe study a multi‐stage dynamic assignment interdiction (DAI) game in which two agents, a user and an attacker, compete in the underlying bipartite assignment graph. The user wishes to assign a set of tasks at the minimum cost, and the attacker seeks to interdict a subset of arcs to maximize the user's objective. The user assigns exactly one task per stage, and the assignment costs and interdiction impacts vary across stages. Before any stage commences in the game, the attacker can interdict arcs subject to a cardinality constraint. An interdicted arc can still be used by the user, but at an increased assignment cost. The goal is to find an optimal sequence of assignments, coupled with the attacker's optimal interdiction strategy. We prove that this problem is strongly NP‐hard, even when the attacker can interdict only one arc. We propose an exact exponential‐state dynamic‐programming algorithm for this problem as well as lower and upper bounds on the optimal objective function value. Our bounds are based on classical interdiction and robust optimization models, and on variations of the DAI game. We examine the efficiency of our algorithms and the quality of our bounds on a set of randomly generated instances. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 373–387, 2017

Solving multi-period interdiction via generalized Bender’s decomposition

This paper considers a novel formulation of the multi-period network interdiction problem. In this model, delivery of the maximum flow as well as the act of interdiction happens over several periods, while the budget of resource for interdiction is limit. It is assumed that when an edge is interdicted in a period, the evader considers a rate of risk of detection at consequent periods. Application of the generalized Benders decomposition algorithm considers solving the resulting mixed-integer nonlinear programming problem. Computational experiences denote reasonable consistency with expectations.

Hardness and approximation for network flow interdiction

In the Network Flow Interdiction problem, an adversary attacks a network in order to minimize the maximum s‐t‐flow. Very little is known about the approximatibility of this problem, despite decades of interest in it. It is, surprisingly, nontrivial to obtain in polynomial time an approximation guarantee that is independent of the number of edges in the graph and their capacities. We present the first such approximation algorithm, which has approximation ratio at most 2(n−1) for any graph with n vertices. We complement the algorithm with a hardness theorem. Past work has shown that Network Flow Interdiction cannot be much easier to approximate than Densest k‐Subgraph. We show that any nϵ‐approximation algorithm for Network Flow Interdiction, or one of several variants, would imply a 2n4ϵ‐approximation algorithm for Densest k‐Subgraph, which is an improvement over past work in terms of the polynomial factor. We also show that Network Flow Interdiction is essentially the same as the Budgeted Minimum s‐t‐Cut problem, and transferring our results gives hardness and an approximation algorithm for that problem, as well. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 378–387 2017

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

AbstractA simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017

Trilevel shortest path network interdiction with partial fortification

This paper presents a new formulation and solution method for partial fortification and interdiction of a trilevel shortest path problem that is a generalization of existing network interdiction models to a more practical setting. We consider a trilevel problem in which a defense plan is determined at the top level to reinforce some network components against a potential attack. The middle-level problem is solved to predict the most damaging potential attack plan to the hardened network, while the bottom-level problem is used to determine the shortest path in the surviving network. In the proposed model, partial interdiction and fortification are permitted to make the model more general and practical. In addition, the costs of interdiction and fortification are minimized in the attacker’s and defender’s objective functions, and the corresponding budget constraints are relaxed.We have devised a trilevel decomposition algorithm to solve the proposed model in a reasonable time. Furthermore, we transform the trilevel partial fortification problem into a bilevel problem using duality and propose a bilevel decomposition algorithm to solve it. We also study three different formulations for a trilevel shortest path interdiction problem with total fortification and solve these problems using a combination of duality, decomposition and relaxation methods. The numerical results show that the optimal objective values obtained by the proposed models are better than those of the total interdiction models. The solution times of the proposed algorithm applied on a set of test problems are significantly less than those of the existing algorithms. This implies that the budget relaxations in the proposed model improve the total traversing cost in the presence of partial attack and fortification.

Two extended formulations for cardinality maximum flow network interdiction problem

We consider the maximum flow network interdiction problem in its cardinality case. There is an integer programming model for this problem by Wood (Math Comput Model 17 (1993), 1–18). Two types of valid inequalities have also been proposed (Altner et al., Oper Res Lett 38 (2010), 33–38 and Wood, Math Comput Model 17 (1993), 1–18) to strengthen the LP relaxation of the integer model. However, due to their combinatorial nature, the number of these inequalities are exponential. Here, we present an equivalent reformulation (extended formulation) for this problem which has a polynomial number of constraints. We also introduce new valid inequalities, and show that the corresponding reformulation of the LP relaxation of the integer model augmented with these inequalities, significantly decreases the integrality gap for a class of network interdiction problems with proven large integrality gaps. Numerical results for some benchmark as well as randomly generated instances are also reported. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 367–377 2017

Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives

Interdiction problems ask about the worst-case impact of a limited change to an underlying optimization problem. They are a natural way to measure the robustness of a system or to identify its weakest spots. Interdiction problems have been studied for a wide variety of classical combinatorial optimization problems. Most interdiction problems are NP-hard, and furthermore, even designing efficient approximation algorithms that allow for estimating the order of magnitude of a worst-case impact has turned out to be very difficult. Not very surprisingly, the few known approximation algorithms are heavily tailored for specific problems. Inspired by previous approaches to network flow interdiction we suggest a general method to obtain pseudoapproximations for many interdiction problems. More precisely, for any α > 0, our algorithm will return either a (1 + α)-approximation or a solution that may overrun the interdiction budget by a factor of at most 1 + 1/α but is also at least as good as the optimal solution that respects the budget. Furthermore, our approach can handle submodular interdiction costs when the underlying problem is to find a maximum weight independent set in a matroid. Additionally, our approach can sometimes be refined by exploiting additional structural properties of the underlying optimization problem to obtain stronger results. We demonstrate this by presenting a polynomial-time approximation scheme for interdicting b-stable sets in bipartite graphs.

A Backward Sampling Framework for Interdiction Problems with Fortification

This paper examines a class of three-stage sequential defender-attacker-defender problems. In these problems the defender first selects a subset of assets to protect, the attacker next damages a subset of unprotected assets in the “interdiction” stage, after which the defender optimizes a “recourse” problem over the surviving assets. These problems are notoriously difficult to optimize, and almost always require the recourse problem to be a convex optimization problem. Our contribution is a new approach to solving defender-attacker-defender problems. We require all variables in the first two stages to be binary-valued, but allow the recourse problem to take any form. The proposed framework focuses on solving the interdiction problem by restricting the defender to select a recourse decision from a sample of feasible vectors. The algorithm then iteratively refines the sample to force finite convergence to an optimal solution. We demonstrate that our algorithm not only solves interdiction problems involving NP-hard recourse problems within reasonable computational limits, but it also solves shortest path fortification and interdiction problems more efficiently than state-of-the-art algorithms tailored for that problem, finding optimal solutions to real-road networks having up to 300,000 nodes and over 1,000,000 arcs.

Simultaneous Convexification of Bilinear Functions over Polytopes with Application to Network Interdiction

We study the simultaneous convexification of graphs of bilinear functions $g^k({x};{y}) = {y}^{\intercal} A^k {x}$ over ${x} \in \ensuremath{\Xi} = \{ {x} \in [0,1]^n \, | \, E{x} \geq {f} \}$ and ${y} \in \Delta_m = \{ {y} \in {R}_+^{m} \, | \, {1^{\intercal}y} \leq 1 \}$. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of integer variables in mixed integer bilinear sets, and to obtain convex hulls of sets with complementarity constraints. Exploiting the structure of $\Xi$, the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings. In particular, we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results.

Assessment of the Electric Grid Interdiction Problem using a nonlinear modeling approach

This paper addresses the vulnerability assessment of power systems under multiple outages, both unintentional and deliberate. In this context, a new model and solution approach for the Electric Grid Interdiction Problem (EGIP) are proposed. The EGIP is based on the interaction of two agents: a Disruptive Agent (DA) that aims at maximizing load shedding by performing attacks in the power system; and a System Operator (SO), who reacts to these attacks by modifying the generation dispatch in order to minimize load shedding. The interaction of these agents is modeled as a bilevel programming problem. Due to its non-convexity and non-linearity the EGIP has been traditionally approached by means of linearized equivalents of the network and recast as a single-level Mixed Integer Linear Programing (MILP) problem. In this paper the EGIP is formulated as a Mixed Integer Non-Linear Programming (MINLP) problem using an AC modeling of the network, providing more accurate results than traditional DC models. Another distinctive modeling feature is introduced: the possibility of attacks not only on branches but also on generators. The proposed model is solved by means of an Iterated Local Search (ILS) algorithm. Performance of the proposed ILS is compared with a conventional Genetic Algorithm (GA). Several tests were performed on two benchmark IEEE test systems showing the applicability and robustness of the proposed approach.

A tri-level covering fortification model for facility protection against disturbance in r-interdiction median problem

The available literature for facility interdiction problems is separately modeled by the two r-interdiction covering and median approaches. In this paper, a tri-level defense facility location model for full coverage in r-interdiction median problem is addressed which considers both modeling approaches, simultaneously. The purpose of this model is to design a proper service system in a way that after a worst case scenario of disturbance, it can utilize its full capacity of providing services. In this regard, we have considered the defense facilities to provide extra protection for service facilities and the purpose is to optimally locate these facilities. The tri-level model is proposed based on leader-follower games as defender-attacker-defender framework. In order to solve the model, three approaches have been used. In the first approach (EX-EX-EX), explicit enumeration method is used for the first and second levels and an exact approach is used for the third level. In the second and third approaches, hybrid methods consisting of genetic algorithm, explicit exact enumeration and exact approach (GA-EX-EX) and biogeography-based algorithm, explicit enumeration method and exact approach (BBO-EX-EX) have been used to tackle the problem in a reasonable time. Finally, the proposed approaches are used to solve 27 random instance problems. Comparing the proposed meta-heuristics and the exact approach and studying the numerical examples solved using these approaches are quite satisfactory.

Controlling Floods by Optimization Methods

Floods as natural phenomena exist and will continue to occur. No manmade project can stop a flood from happening, but there are several effective methods to reduce its risk, aftermath and consequences. This paper considers a novel formulation of the network interdiction problem with application on controlling floods. The aim of this work is to identify those arcs prone for flood, and implementing flood-control projects on these arcs, while required budget for deriving all these operations is bounded. the model is tested on a real-world case in Tabriz-Iran, and results are visualized.

Risk-Averse Shortest Path Interdiction

We consider a Stackelberg game in a network where a leader minimizes the cost of interdicting arcs and a follower seeks the shortest distance between given origin and destination nodes under uncertain arc traveling cost. In particular, we consider a risk-averse leader, who aims to keep high probability that the follower’s traveling distance is longer than a given threshold, interpreted by a chance constraint. Under the assumption of a wait-and-see follower—i.e., the follower selects a shortest path after seeing realizations of the random arc cost—we propose a branch-and-cut algorithm and apply lifting techniques to exploit the combinatorial structure of the risk-averse leader’s interdiction problem. We demonstrate the computational efficacy of our approaches, risk-averse interdiction solution patterns, and result sensitivity, by testing instances of randomly generated grid networks and real-world transportation networks.

Parametric multiroute flow and its application to multilink-attack network

We investigate variants of the max-flow problem in a network under k attacks. The network interdiction problem is to find the minimum max-flow value among (mk) networks that can be obtained by deleting each set of k links. The adaptive network flow problem is to find a flow of the network such that the flow value is maximum against any set of k links attack, when deleting the corresponding flow to those k links in the original flow. First, we prove that max-(k+1)-route flow is a (k+1)-approximation for both problems. Also, we develop a polynomial-time heuristic algorithm for both cases, called the iterative multiroute flow. Then in a second phase, we investigate properties of the function taking the real value h to the max h-route flow value, and apply the result to solve both of the problems. We show that the function is piecewise hyperbolic, and modify a standard parametric optimization technique to find this function. The running time of the algorithm is O(λT), when λ is a source–sink edge connectivity of our network and T the computation time of a max-flow algorithm. We show that for some instances, when h is optimally chosen, the max- h-route flow is an exact solution for both problems.

An Extended Network Interdiction Problem for Optimal Toll Control

We study an extension of the shortest path network interdiction problem and present a novel real-world application in this area. We consider the problem of determining optimal locations for toll control stations on the arcs of a transportation network. We handle the fact that drivers can avoid control stations on parallel secondary roads. The problem is formulated as a mixed integer program and solved using Benders decomposition. We present experimental results for the application of our models to German motorways.